[{"Name":"The Concept of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Function","Duration":"8m 56s","ChapterTopicVideoID":6243,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this clip, we\u0027ll learn what a function is in mathematics,"},{"Start":"00:04.200 ","End":"00:05.940","Text":"not in everyday life."},{"Start":"00:05.940 ","End":"00:10.320","Text":"To help you understand the meaning of a function in mathematics,"},{"Start":"00:10.320 ","End":"00:12.735","Text":"we\u0027ll sketch a little house here."},{"Start":"00:12.735 ","End":"00:15.000","Text":"This is the base of the house."},{"Start":"00:15.000 ","End":"00:18.705","Text":"It\u0027s a square with side of length x,"},{"Start":"00:18.705 ","End":"00:20.865","Text":"that\u0027s x, and this is x."},{"Start":"00:20.865 ","End":"00:23.940","Text":"The house has a roof in a shape of a triangle."},{"Start":"00:23.940 ","End":"00:29.085","Text":"I know that the height of this triangle of the roof is 4."},{"Start":"00:29.085 ","End":"00:31.470","Text":"The house has an area,"},{"Start":"00:31.470 ","End":"00:36.135","Text":"let y denote the area of the house."},{"Start":"00:36.135 ","End":"00:37.830","Text":"As we said already,"},{"Start":"00:37.830 ","End":"00:44.320","Text":"x denotes the length of the side of the base of the house."},{"Start":"00:44.320 ","End":"00:49.415","Text":"y is the area of the whole house which is made up of 2 parts."},{"Start":"00:49.415 ","End":"00:55.585","Text":"There\u0027s the base here and there\u0027s the triangular roof."},{"Start":"00:55.585 ","End":"01:05.120","Text":"I can write a formula for y. I can say that y is equal to the area of this square base,"},{"Start":"01:05.120 ","End":"01:06.935","Text":"which is x times x,"},{"Start":"01:06.935 ","End":"01:09.065","Text":"which is x squared."},{"Start":"01:09.065 ","End":"01:11.390","Text":"That\u0027s the base. Then there\u0027s the roof,"},{"Start":"01:11.390 ","End":"01:12.935","Text":"which is a triangle."},{"Start":"01:12.935 ","End":"01:18.245","Text":"The formula for the area of a triangle is 1/2 base times height."},{"Start":"01:18.245 ","End":"01:22.490","Text":"We get 1/2 times x."},{"Start":"01:22.490 ","End":"01:24.740","Text":"The base of the triangle is also x,"},{"Start":"01:24.740 ","End":"01:28.090","Text":"times the height, which is 4."},{"Start":"01:28.090 ","End":"01:31.695","Text":"1/2 times x times 4 is simply 2x."},{"Start":"01:31.695 ","End":"01:35.025","Text":"I\u0027ve expressed y in terms of x."},{"Start":"01:35.025 ","End":"01:38.180","Text":"In this case, y depends on x."},{"Start":"01:38.180 ","End":"01:43.865","Text":"Each value of x determines the corresponding value of y."},{"Start":"01:43.865 ","End":"01:45.890","Text":"Let\u0027s see how this works."},{"Start":"01:45.890 ","End":"01:49.130","Text":"If x is equal to 1,"},{"Start":"01:49.130 ","End":"01:52.190","Text":"then y will equal from this formula,"},{"Start":"01:52.190 ","End":"01:56.635","Text":"1 squared plus twice 1 will be 3."},{"Start":"01:56.635 ","End":"02:01.385","Text":"In other words, if the base of the house has a side of length 1,"},{"Start":"02:01.385 ","End":"02:05.580","Text":"the area of the whole house will be 3."},{"Start":"02:05.680 ","End":"02:10.705","Text":"If for example x is equal to 2,"},{"Start":"02:10.705 ","End":"02:18.295","Text":"then y equals 2 squared plus twice 2 is 8, 4 plus 4."},{"Start":"02:18.295 ","End":"02:22.195","Text":"If the house has base of length 2,"},{"Start":"02:22.195 ","End":"02:24.665","Text":"the area of the house will be 8."},{"Start":"02:24.665 ","End":"02:30.685","Text":"Similarly, if say x is equal to 5,"},{"Start":"02:30.685 ","End":"02:35.230","Text":"then we\u0027ll get y equals 5 squared is 25 plus twice 5, it\u0027s a 10,"},{"Start":"02:35.230 ","End":"02:40.245","Text":"then y is equal to 35, and so on."},{"Start":"02:40.245 ","End":"02:44.735","Text":"We see that y depends on x."},{"Start":"02:44.735 ","End":"02:52.130","Text":"That is, the variable y depends on the variable x in such a way that to"},{"Start":"02:52.130 ","End":"02:59.420","Text":"every value of x there corresponds exactly 1 value of y. I\u0027ll write this down shortly."},{"Start":"02:59.420 ","End":"03:04.340","Text":"But first I\u0027d like to illustrate with the help of a couple of circles,"},{"Start":"03:04.340 ","End":"03:06.885","Text":"well, more ovals and circles."},{"Start":"03:06.885 ","End":"03:11.060","Text":"This 1 represents the values of x,"},{"Start":"03:11.060 ","End":"03:14.540","Text":"and this 1 represents the values of y."},{"Start":"03:14.540 ","End":"03:17.990","Text":"The value of x, which is 1,"},{"Start":"03:17.990 ","End":"03:23.150","Text":"has a corresponding value of y, which is 3."},{"Start":"03:23.150 ","End":"03:25.510","Text":"The x that is,"},{"Start":"03:25.510 ","End":"03:31.250","Text":"2 get the corresponding y, which is 8,"},{"Start":"03:31.250 ","End":"03:33.875","Text":"the x that is 5,"},{"Start":"03:33.875 ","End":"03:36.950","Text":"is going to get a corresponding y,"},{"Start":"03:36.950 ","End":"03:40.425","Text":"which is 35, and so on."},{"Start":"03:40.425 ","End":"03:42.705","Text":"To each value of x,"},{"Start":"03:42.705 ","End":"03:47.135","Text":"there corresponds exactly 1 value of y."},{"Start":"03:47.135 ","End":"03:51.050","Text":"This brings us to a formal definition of function,"},{"Start":"03:51.050 ","End":"03:56.030","Text":"which I\u0027ll say out loud first and I\u0027ll write it down. It goes as follows."},{"Start":"03:56.030 ","End":"04:01.099","Text":"If a variable y depends on the variable x,"},{"Start":"04:01.099 ","End":"04:02.615","Text":"like we see here,"},{"Start":"04:02.615 ","End":"04:10.085","Text":"in such a way that for each value of x there corresponds exactly 1 value of y,"},{"Start":"04:10.085 ","End":"04:14.180","Text":"then we say that y is a function of x."},{"Start":"04:14.180 ","End":"04:20.785","Text":"In this case, the area of the house is a function of the length of the side of the base."},{"Start":"04:20.785 ","End":"04:23.614","Text":"Let\u0027s write this down properly."},{"Start":"04:23.614 ","End":"04:28.760","Text":"A variable y is called a function of"},{"Start":"04:28.760 ","End":"04:35.990","Text":"another variable x if y depends on x in such a way that for each value of x,"},{"Start":"04:35.990 ","End":"04:41.485","Text":"there corresponds exactly 1 value of y."},{"Start":"04:41.485 ","End":"04:45.710","Text":"Meanwhile, we\u0027ve had just 1 example of a function."},{"Start":"04:45.710 ","End":"04:48.305","Text":"This was the function that we had."},{"Start":"04:48.305 ","End":"04:52.460","Text":"It came a little problem about a house and"},{"Start":"04:52.460 ","End":"04:56.555","Text":"the length of the side of the base and the area of the house."},{"Start":"04:56.555 ","End":"04:59.810","Text":"But a function doesn\u0027t have to come out of a story."},{"Start":"04:59.810 ","End":"05:02.300","Text":"It could just be totally abstract."},{"Start":"05:02.300 ","End":"05:11.855","Text":"I could write y equals x plus 1 over x minus 1,"},{"Start":"05:11.855 ","End":"05:17.960","Text":"totally abstractly, and that would be a function because for each x,"},{"Start":"05:17.960 ","End":"05:20.915","Text":"for example, x equals 0,"},{"Start":"05:20.915 ","End":"05:24.440","Text":"I can get exactly 1 value of y."},{"Start":"05:24.440 ","End":"05:30.855","Text":"In this case, 0 plus 1 over 0 minus 1 is minus 1."},{"Start":"05:30.855 ","End":"05:34.070","Text":"Or if x equals 2, another example,"},{"Start":"05:34.070 ","End":"05:38.645","Text":"I would get y equals 2 plus 1 over 2 minus 1,"},{"Start":"05:38.645 ","End":"05:43.535","Text":"which is 3 over 1, which would be 3, and so on."},{"Start":"05:43.535 ","End":"05:46.445","Text":"What I would like to point out is that"},{"Start":"05:46.445 ","End":"05:51.110","Text":"not every time that you have a dependency between y and x,"},{"Start":"05:51.110 ","End":"05:53.990","Text":"and even an equation involving y and x,"},{"Start":"05:53.990 ","End":"05:57.140","Text":"does it mean that y is a function of x?"},{"Start":"05:57.140 ","End":"06:00.005","Text":"I want to look at the following example."},{"Start":"06:00.005 ","End":"06:05.240","Text":"Let\u0027s take y squared equals x."},{"Start":"06:05.240 ","End":"06:08.330","Text":"Certainly y is dependent on x,"},{"Start":"06:08.330 ","End":"06:10.820","Text":"but is y a function of x?"},{"Start":"06:10.820 ","End":"06:15.850","Text":"Let\u0027s take, for instance, x equals 4."},{"Start":"06:15.850 ","End":"06:21.240","Text":"What is y? We have that y squared equals 4."},{"Start":"06:21.240 ","End":"06:23.485","Text":"But for y squared equals 4,"},{"Start":"06:23.485 ","End":"06:25.245","Text":"there are 2 solutions."},{"Start":"06:25.245 ","End":"06:29.735","Text":"We could have y equals 2 and that would satisfy the equation."},{"Start":"06:29.735 ","End":"06:36.685","Text":"But we could also have y equals minus 2 and that would also satisfy the equation."},{"Start":"06:36.685 ","End":"06:41.780","Text":"1 value of x gives 2 values of y,"},{"Start":"06:41.780 ","End":"06:46.280","Text":"which contradicts the exactly 1 that is written here."},{"Start":"06:46.280 ","End":"06:49.400","Text":"In fact, I can show you another bad thing."},{"Start":"06:49.400 ","End":"06:54.545","Text":"Suppose I took x equals minus 9,"},{"Start":"06:54.545 ","End":"06:56.960","Text":"then what is y equal?"},{"Start":"06:56.960 ","End":"07:01.040","Text":"Well, I have to have y squared is minus 9,"},{"Start":"07:01.040 ","End":"07:06.140","Text":"but the square of a number can never be negative that there is no such y."},{"Start":"07:06.140 ","End":"07:09.350","Text":"For x equals 4, we have 2 values of y."},{"Start":"07:09.350 ","End":"07:10.850","Text":"For x equals minus 9,"},{"Start":"07:10.850 ","End":"07:13.040","Text":"we have no values of y."},{"Start":"07:13.040 ","End":"07:18.530","Text":"It\u0027s important that there be exactly 1 in order for it to be a function."},{"Start":"07:18.530 ","End":"07:22.220","Text":"Let\u0027s return a minute to the first example,"},{"Start":"07:22.220 ","End":"07:24.045","Text":"the 1 with the house."},{"Start":"07:24.045 ","End":"07:26.840","Text":"I\u0027d like to just take another couple of values here."},{"Start":"07:26.840 ","End":"07:29.285","Text":"We did x equals 1, 2, and 5."},{"Start":"07:29.285 ","End":"07:33.880","Text":"Let\u0027s see what happens if x equals 0."},{"Start":"07:33.880 ","End":"07:35.750","Text":"I\u0027m like continuing."},{"Start":"07:35.750 ","End":"07:41.165","Text":"Then y equals 0 squared plus twice 0 is 0."},{"Start":"07:41.165 ","End":"07:46.285","Text":"What happens if x equals minus 2?"},{"Start":"07:46.285 ","End":"07:52.130","Text":"Substituting here, minus 2 squared is 4 twice minus 2 is minus 4."},{"Start":"07:52.130 ","End":"07:56.794","Text":"4 minus 4 means that y is also 0."},{"Start":"07:56.794 ","End":"07:59.360","Text":"My question is, is this a problem?"},{"Start":"07:59.360 ","End":"08:02.105","Text":"The answer is no, it isn\u0027t."},{"Start":"08:02.105 ","End":"08:06.040","Text":"Let me illustrate this in this diagram below."},{"Start":"08:06.040 ","End":"08:13.970","Text":"We have here the point where x is 0 and here the point where x is minus 2,"},{"Start":"08:13.970 ","End":"08:19.114","Text":"and both of them go to the same value of y, which is 0."},{"Start":"08:19.114 ","End":"08:22.130","Text":"But this is not a problem because"},{"Start":"08:22.130 ","End":"08:27.935","Text":"our requirement is that each x should have just 1 value of y."},{"Start":"08:27.935 ","End":"08:34.415","Text":"It\u0027s possible that a single value of y can have more than 1 value of x."},{"Start":"08:34.415 ","End":"08:36.995","Text":"That is not considered a problem."},{"Start":"08:36.995 ","End":"08:43.670","Text":"Let\u0027s contrast this with the previous example of y squared equals x,"},{"Start":"08:43.670 ","End":"08:47.540","Text":"where we had a single x going to 2 different y\u0027s."},{"Start":"08:47.540 ","End":"08:49.790","Text":"This is not allowed,"},{"Start":"08:49.790 ","End":"08:55.620","Text":"and therefore this 1 is not a function."}],"ID":6255},{"Watched":false,"Name":"Notation for Functions","Duration":"7m 25s","ChapterTopicVideoID":6244,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.365","Text":"In this clip, I\u0027d like to show you an additional notation for functions."},{"Start":"00:04.365 ","End":"00:07.140","Text":"The old notation would go something like this."},{"Start":"00:07.140 ","End":"00:17.160","Text":"We would say y equals x squared plus 4x plus 1 and then y would be a function of x."},{"Start":"00:17.160 ","End":"00:18.705","Text":"In the new notation,"},{"Start":"00:18.705 ","End":"00:22.665","Text":"we will write something like f of x,"},{"Start":"00:22.665 ","End":"00:23.820","Text":"that\u0027s how it\u0027s pronounced,"},{"Start":"00:23.820 ","End":"00:25.050","Text":"we say the word of,"},{"Start":"00:25.050 ","End":"00:31.050","Text":"f of x is equal to x squared plus 4x plus 1."},{"Start":"00:31.050 ","End":"00:33.885","Text":"For example, in the new notation,"},{"Start":"00:33.885 ","End":"00:37.010","Text":"I would say f of 4 equals,"},{"Start":"00:37.010 ","End":"00:39.230","Text":"and I would do the same thing I would do if I was"},{"Start":"00:39.230 ","End":"00:42.290","Text":"substituting x equals 4 for figuring out y. I would"},{"Start":"00:42.290 ","End":"00:50.525","Text":"say 4 squared is 16 plus 4 times 4 is another 16 plus 1 is 33."},{"Start":"00:50.525 ","End":"00:52.865","Text":"F of 4 is 33,"},{"Start":"00:52.865 ","End":"00:58.115","Text":"and we would say something like the value of"},{"Start":"00:58.115 ","End":"01:07.110","Text":"the function f at the point x equals 4 is 33."},{"Start":"01:07.110 ","End":"01:08.420","Text":"In the old notation,"},{"Start":"01:08.420 ","End":"01:14.315","Text":"I might say something like if x equals 4,"},{"Start":"01:14.315 ","End":"01:18.575","Text":"then y equals 33."},{"Start":"01:18.575 ","End":"01:24.170","Text":"We call this the old notation and I call this 1 the new notation,"},{"Start":"01:24.170 ","End":"01:26.839","Text":"but actually both are used."},{"Start":"01:26.839 ","End":"01:29.620","Text":"It\u0027s just that for you, this is newer."},{"Start":"01:29.620 ","End":"01:32.615","Text":"Once again, for the new notation,"},{"Start":"01:32.615 ","End":"01:36.110","Text":"we write f of 4 equals 33."},{"Start":"01:36.110 ","End":"01:40.430","Text":"In longhand we would say the value of the function f,"},{"Start":"01:40.430 ","End":"01:42.360","Text":"f stands for function,"},{"Start":"01:42.360 ","End":"01:45.750","Text":"at the point x is 4,"},{"Start":"01:45.750 ","End":"01:48.420","Text":"x equals 4 is 33."},{"Start":"01:48.420 ","End":"01:52.895","Text":"When we say the value of the function, we mean the y."},{"Start":"01:52.895 ","End":"01:56.960","Text":"If I want to know what is f of 0,"},{"Start":"01:56.960 ","End":"01:59.040","Text":"then in the old notation,"},{"Start":"01:59.040 ","End":"02:02.690","Text":"it\u0027s like asking what is y when x is 0,"},{"Start":"02:02.690 ","End":"02:06.775","Text":"and you substitute 0, we quickly see that this equal to 1."},{"Start":"02:06.775 ","End":"02:11.030","Text":"The old way I would say when x is 0, y is 1."},{"Start":"02:11.030 ","End":"02:16.910","Text":"Now I would say the value of the function f at the point x equals 0 is 1."},{"Start":"02:16.910 ","End":"02:20.465","Text":"This new notation has some advantages."},{"Start":"02:20.465 ","End":"02:23.810","Text":"When I write f of x, I know that it\u0027s x,"},{"Start":"02:23.810 ","End":"02:27.335","Text":"which is the variable because there are situations where it\u0027s not clear."},{"Start":"02:27.335 ","End":"02:31.985","Text":"The other thing is that function names, in this case,"},{"Start":"02:31.985 ","End":"02:37.700","Text":"f, can be several letters long, unlike regular variables."},{"Start":"02:37.700 ","End":"02:40.260","Text":"For example, in economics,"},{"Start":"02:40.260 ","End":"02:43.420","Text":"common function name is TC,"},{"Start":"02:43.420 ","End":"02:46.380","Text":"which stands for total cost,"},{"Start":"02:46.380 ","End":"02:49.085","Text":"and everyone knows what is TC."},{"Start":"02:49.085 ","End":"02:51.715","Text":"There are similarly names used in physics and"},{"Start":"02:51.715 ","End":"02:55.630","Text":"other branches that are familiar and they can be descriptive."},{"Start":"02:55.630 ","End":"03:03.655","Text":"In economics, TC stands for total cost, that\u0027s in economics."},{"Start":"03:03.655 ","End":"03:08.200","Text":"In mathematics, common function names are f,"},{"Start":"03:08.200 ","End":"03:12.420","Text":"g, h. Anyway not to get worried about that,"},{"Start":"03:12.420 ","End":"03:14.650","Text":"the point is that a function can have a name,"},{"Start":"03:14.650 ","End":"03:17.470","Text":"doesn\u0027t have to be f. The next example I\u0027m"},{"Start":"03:17.470 ","End":"03:20.845","Text":"about to give will be from economics and we\u0027ll use the TC,"},{"Start":"03:20.845 ","End":"03:23.245","Text":"which is total cost function."},{"Start":"03:23.245 ","End":"03:29.770","Text":"In this example, we\u0027re given that Danny bought some candy bars."},{"Start":"03:29.770 ","End":"03:38.820","Text":"He bought x Mars bars and he bought x squared KitKats."},{"Start":"03:38.820 ","End":"03:40.530","Text":"I added some illustration here,"},{"Start":"03:40.530 ","End":"03:42.570","Text":"I\u0027m not advertising these products."},{"Start":"03:42.570 ","End":"03:46.075","Text":"Anyway, the cost of each,"},{"Start":"03:46.075 ","End":"03:47.230","Text":"I\u0027ll tell you what it is."},{"Start":"03:47.230 ","End":"03:52.785","Text":"The cost of a Mars is"},{"Start":"03:52.785 ","End":"04:00.710","Text":"40 cents and a KitKat costs 50 cents."},{"Start":"04:00.710 ","End":"04:04.149","Text":"I\u0027d like to explain about the x and the x squared."},{"Start":"04:04.149 ","End":"04:10.815","Text":"X is a variable representing how many Mars bars Danny bought."},{"Start":"04:10.815 ","End":"04:15.140","Text":"X could be anything but it has to be a positive whole number of course."},{"Start":"04:15.140 ","End":"04:17.635","Text":"Like if x is 3,"},{"Start":"04:17.635 ","End":"04:21.255","Text":"and he bought 3 Mars bars and x squared is 9,"},{"Start":"04:21.255 ","End":"04:23.145","Text":"so he bought 9 KitKats."},{"Start":"04:23.145 ","End":"04:24.270","Text":"If x is 2,"},{"Start":"04:24.270 ","End":"04:27.660","Text":"then he brought 2 Mars bars and 4 KitKats."},{"Start":"04:27.660 ","End":"04:32.550","Text":"If x is 5, 5 Mars bars and 5 squared is 25 KitKats and so on."},{"Start":"04:32.550 ","End":"04:35.555","Text":"The question part, there\u0027s 2 parts."},{"Start":"04:35.555 ","End":"04:46.560","Text":"A, write the cost function or the total cost function for Danny\u0027s expenses."},{"Start":"04:46.670 ","End":"04:52.835","Text":"Then there\u0027s part B, where you got to figure out how much did"},{"Start":"04:52.835 ","End":"04:59.965","Text":"Danny spend if he bought 4 Mars bars?"},{"Start":"04:59.965 ","End":"05:05.150","Text":"We need to express the cost as a function of x,"},{"Start":"05:05.150 ","End":"05:06.710","Text":"the number of Mars bars."},{"Start":"05:06.710 ","End":"05:08.480","Text":"Let\u0027s give the function a name,"},{"Start":"05:08.480 ","End":"05:11.440","Text":"let\u0027s call it TC."},{"Start":"05:11.440 ","End":"05:18.540","Text":"We get TC and it\u0027s a function of x and what it\u0027s going to equal? Let\u0027s do the math."},{"Start":"05:18.540 ","End":"05:25.875","Text":"If each Mars bar costs 40 cents and he bought x of them,"},{"Start":"05:25.875 ","End":"05:30.930","Text":"so this is equal to 40x if we work in cents."},{"Start":"05:30.930 ","End":"05:33.165","Text":"But I would prefer to work in dollars,"},{"Start":"05:33.165 ","End":"05:36.460","Text":"so this 40 is a $0.4."},{"Start":"05:37.730 ","End":"05:42.030","Text":"This is the part of the Mars bars that he also bought KitKat."},{"Start":"05:42.030 ","End":"05:43.635","Text":"How many of these did he buy?"},{"Start":"05:43.635 ","End":"05:45.705","Text":"He bought x squared KitKats,"},{"Start":"05:45.705 ","End":"05:49.800","Text":"and each of them costs 50 cents or $0.5,"},{"Start":"05:49.800 ","End":"05:53.215","Text":"so we\u0027ve got an extra 0.5x"},{"Start":"05:53.215 ","End":"05:58.250","Text":"squared number of items times the cost per item for the KitKat."},{"Start":"05:58.250 ","End":"06:03.295","Text":"Altogether, this is the total cost of his candy bar expenses."},{"Start":"06:03.295 ","End":"06:07.050","Text":"This is in fact the answer to part A."},{"Start":"06:07.050 ","End":"06:09.330","Text":"Now onto part B,"},{"Start":"06:09.330 ","End":"06:12.895","Text":"how much did Danny spend if he bought 4 Mars bars?"},{"Start":"06:12.895 ","End":"06:15.800","Text":"That must mean that x is equal to 4."},{"Start":"06:15.800 ","End":"06:20.840","Text":"This means that we need TC of 4 and this"},{"Start":"06:20.840 ","End":"06:25.755","Text":"is going to equal just simply substituting 4 instead of x,"},{"Start":"06:25.755 ","End":"06:35.730","Text":"so we get 0.4 times 4 plus 0.5 times 4 squared."},{"Start":"06:35.730 ","End":"06:37.305","Text":"What does this equal?"},{"Start":"06:37.305 ","End":"06:42.585","Text":"4 times 0.4 is 1.6,"},{"Start":"06:42.585 ","End":"06:47.310","Text":"4 squared is 16 and"},{"Start":"06:47.310 ","End":"06:53.235","Text":"16 times 0.5 is 8 because 0.5 is 1/2."},{"Start":"06:53.235 ","End":"06:58.140","Text":"Altogether we have got 9.6."},{"Start":"06:58.140 ","End":"07:01.130","Text":"I\u0027m just reminding you that it\u0027s in dollars and"},{"Start":"07:01.130 ","End":"07:04.655","Text":"some people like to write the extra decimal place."},{"Start":"07:04.655 ","End":"07:14.145","Text":"This is the answer to part B and the cost function here is the answer to part A."},{"Start":"07:14.145 ","End":"07:17.990","Text":"This little example of using a meaningful function name,"},{"Start":"07:17.990 ","End":"07:20.870","Text":"in this case, TC from the world of economics,"},{"Start":"07:20.870 ","End":"07:22.940","Text":"where it means total cost."},{"Start":"07:22.940 ","End":"07:25.890","Text":"I\u0027m done with this clip."}],"ID":6256},{"Watched":false,"Name":"The Domain of Definition of a Function","Duration":"5m 53s","ChapterTopicVideoID":30337,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this clip, we\u0027ll talk about the domain of definition of a function."},{"Start":"00:04.680 ","End":"00:08.460","Text":"Domain of definition sometimes is just called the domain."},{"Start":"00:08.460 ","End":"00:09.930","Text":"What does it mean?"},{"Start":"00:09.930 ","End":"00:12.150","Text":"Basically, when we have a function,"},{"Start":"00:12.150 ","End":"00:16.845","Text":"the domain of definition simply means the values of x,"},{"Start":"00:16.845 ","End":"00:20.685","Text":"it\u0027s actually a set of values but just called the the values of x,"},{"Start":"00:20.685 ","End":"00:22.941","Text":"for which makes sense to say f(x),"},{"Start":"00:22.941 ","End":"00:25.724","Text":"or if we put it, in other words,"},{"Start":"00:25.724 ","End":"00:30.390","Text":"which values of x are we allowed to substitute in the function f?"},{"Start":"00:30.390 ","End":"00:32.085","Text":"Let\u0027s take an example."},{"Start":"00:32.085 ","End":"00:42.315","Text":"The first example we will take f(x) is 24 over 2 minus x."},{"Start":"00:42.315 ","End":"00:47.450","Text":"Let\u0027s see what values of x makes sense to substitute here."},{"Start":"00:47.450 ","End":"00:51.175","Text":"For example, could we put x equals 0?"},{"Start":"00:51.175 ","End":"00:55.785","Text":"Certainly, 2 minus 0 is 2,"},{"Start":"00:55.785 ","End":"00:57.345","Text":"24 over 2 is 12."},{"Start":"00:57.345 ","End":"00:59.760","Text":"Could we put x equals 1?"},{"Start":"00:59.760 ","End":"01:04.110","Text":"Yeah, 24 over 2 minus 1 is 24."},{"Start":"01:04.110 ","End":"01:06.935","Text":"We put x equals 3,"},{"Start":"01:06.935 ","End":"01:09.215","Text":"2 minus 3 is negative 1,"},{"Start":"01:09.215 ","End":"01:12.305","Text":"24 over negative 1 is minus 24."},{"Start":"01:12.305 ","End":"01:14.900","Text":"So what can we put?"},{"Start":"01:14.900 ","End":"01:20.345","Text":"If you remember, we\u0027re not allowed to divide by 0,"},{"Start":"01:20.345 ","End":"01:23.510","Text":"that\u0027s the only number we\u0027re not allowed to divide by."},{"Start":"01:23.510 ","End":"01:26.620","Text":"What happens if we put x equals 2?"},{"Start":"01:26.620 ","End":"01:28.725","Text":"If we put x equals 2,"},{"Start":"01:28.725 ","End":"01:31.050","Text":"we\u0027ll get 24 over 0,"},{"Start":"01:31.050 ","End":"01:34.150","Text":"and we\u0027re not allowed to divide by 0."},{"Start":"01:34.150 ","End":"01:36.360","Text":"If you look at it closely,"},{"Start":"01:36.360 ","End":"01:38.445","Text":"that\u0027s the only problem that can be solved."},{"Start":"01:38.445 ","End":"01:42.865","Text":"Every value of x is okay except for x equals 2."},{"Start":"01:42.865 ","End":"01:47.330","Text":"We say that the domain of definition is everything except"},{"Start":"01:47.330 ","End":"01:52.055","Text":"2 which we can write simply as x not equal to 2."},{"Start":"01:52.055 ","End":"01:55.045","Text":"Now let\u0027s take another example."},{"Start":"01:55.045 ","End":"01:56.520","Text":"As our next example,"},{"Start":"01:56.520 ","End":"02:04.260","Text":"we\u0027ll have f(x) is equal to the square root of x minus 1."},{"Start":"02:04.260 ","End":"02:06.940","Text":"We start looking at values of x,"},{"Start":"02:06.940 ","End":"02:08.725","Text":"which we could put in here."},{"Start":"02:08.725 ","End":"02:13.400","Text":"We try let\u0027s say x equals 2,"},{"Start":"02:13.400 ","End":"02:16.800","Text":"and you get 2 minus 1 is 1,"},{"Start":"02:16.800 ","End":"02:19.635","Text":"square root of 1, that\u0027s fine,"},{"Start":"02:19.635 ","End":"02:22.930","Text":"and we could take x equals 5,"},{"Start":"02:22.930 ","End":"02:26.680","Text":"5 minus 1 is 4, no problem."},{"Start":"02:26.680 ","End":"02:29.575","Text":"You could even take x equals 3."},{"Start":"02:29.575 ","End":"02:32.365","Text":"3 minus 1 is 2,"},{"Start":"02:32.365 ","End":"02:33.550","Text":"square root of 2,"},{"Start":"02:33.550 ","End":"02:34.660","Text":"which is not a whole number,"},{"Start":"02:34.660 ","End":"02:37.065","Text":"but I can say square root of 2."},{"Start":"02:37.065 ","End":"02:41.630","Text":"X equals a 100, square root of 99."},{"Start":"02:41.630 ","End":"02:44.580","Text":"Let\u0027s try x equals 0,"},{"Start":"02:44.580 ","End":"02:49.630","Text":"square root of minus 1."},{"Start":"02:49.630 ","End":"02:52.410","Text":"Seem to remember that square roots,"},{"Start":"02:52.410 ","End":"02:56.210","Text":"you can only take for positive numbers and 0,"},{"Start":"02:56.210 ","End":"02:58.100","Text":"but you can\u0027t take for negative numbers."},{"Start":"02:58.100 ","End":"02:59.360","Text":"If x is 1,"},{"Start":"02:59.360 ","End":"03:00.890","Text":"square root of 1 minus 1,"},{"Start":"03:00.890 ","End":"03:03.095","Text":"square root of 0 is 0 is okay,"},{"Start":"03:03.095 ","End":"03:05.120","Text":"x is 1 is okay,"},{"Start":"03:05.120 ","End":"03:07.315","Text":"x is 0 is not okay,"},{"Start":"03:07.315 ","End":"03:09.270","Text":"x is 2 is okay."},{"Start":"03:09.270 ","End":"03:11.495","Text":"In general, if you remember,"},{"Start":"03:11.495 ","End":"03:14.105","Text":"when can we take the square root of something,"},{"Start":"03:14.105 ","End":"03:19.430","Text":"which I\u0027ll symbolically just write as a little box,"},{"Start":"03:19.430 ","End":"03:23.520","Text":"to take the square root of x or when does it make sense,"},{"Start":"03:23.520 ","End":"03:26.525","Text":"it\u0027s when we have a positive number."},{"Start":"03:26.525 ","End":"03:29.630","Text":"But as I said, not just positive, we could even have 0."},{"Start":"03:29.630 ","End":"03:31.445","Text":"As long it\u0027s not negative,"},{"Start":"03:31.445 ","End":"03:35.635","Text":"usually called sometimes non-negative numbers when it\u0027s bigger or equal to."},{"Start":"03:35.635 ","End":"03:40.410","Text":"In this case, x minus 1 has to be non-negative."},{"Start":"03:40.410 ","End":"03:46.220","Text":"The domain of definition is x bigger or equal to"},{"Start":"03:46.220 ","End":"03:52.565","Text":"1 because that will give us x minus 1 is bigger or equal to 0 as here."},{"Start":"03:52.565 ","End":"03:54.770","Text":"There\u0027s another example."},{"Start":"03:54.770 ","End":"03:56.569","Text":"Let us take a third example."},{"Start":"03:56.569 ","End":"03:58.445","Text":"In this third example,"},{"Start":"03:58.445 ","End":"04:03.410","Text":"I\u0027m going to take the story from the previous clip,"},{"Start":"04:03.410 ","End":"04:05.210","Text":"and just in case you missed it,"},{"Start":"04:05.210 ","End":"04:08.595","Text":"that\u0027s where we drew a little house."},{"Start":"04:08.595 ","End":"04:12.620","Text":"At the end we made a computation and f(x) was the area."},{"Start":"04:12.620 ","End":"04:14.495","Text":"We actually wrote the formula,"},{"Start":"04:14.495 ","End":"04:16.130","Text":"y was over there,"},{"Start":"04:16.130 ","End":"04:18.515","Text":"or f(x), x squared plus 2 x."},{"Start":"04:18.515 ","End":"04:20.525","Text":"You might be tempted to say,"},{"Start":"04:20.525 ","End":"04:22.895","Text":"when we looking at the domain of definition,"},{"Start":"04:22.895 ","End":"04:25.985","Text":"that you\u0027re allowed to substitute every value of x."},{"Start":"04:25.985 ","End":"04:28.565","Text":"There is nothing out of bounds."},{"Start":"04:28.565 ","End":"04:31.820","Text":"Any number positive or negative, or whatever, we certainly,"},{"Start":"04:31.820 ","End":"04:38.395","Text":"we can square it and take twice of it and add but if you wrote all x,"},{"Start":"04:38.395 ","End":"04:41.095","Text":"I\u0027m putting a big question mark here."},{"Start":"04:41.095 ","End":"04:43.120","Text":"If you wrote that, you would be wrong"},{"Start":"04:43.120 ","End":"04:47.440","Text":"because this is not a purely mathematical abstract problem,"},{"Start":"04:47.440 ","End":"04:53.310","Text":"this comes from the real-world and here x represents the length of the base of the house."},{"Start":"04:53.310 ","End":"04:56.860","Text":"If x is a length of some part of the house,"},{"Start":"04:56.860 ","End":"04:58.615","Text":"then x can\u0027t be everything."},{"Start":"04:58.615 ","End":"05:01.000","Text":"X has to be strictly positive,"},{"Start":"05:01.000 ","End":"05:03.205","Text":"so the answer is no."},{"Start":"05:03.205 ","End":"05:11.485","Text":"The answer is that x is bigger than 0 because x isn\u0027t abstract,"},{"Start":"05:11.485 ","End":"05:14.934","Text":"it represents a length."},{"Start":"05:14.934 ","End":"05:17.970","Text":"This can happen often,"},{"Start":"05:17.970 ","End":"05:23.030","Text":"but mathematically there is no restriction but because of the nature of the variable,"},{"Start":"05:23.030 ","End":"05:24.469","Text":"it may be a length,"},{"Start":"05:24.469 ","End":"05:26.150","Text":"it may be a weight,"},{"Start":"05:26.150 ","End":"05:29.660","Text":"which case may be is 0 is allowed or it could even"},{"Start":"05:29.660 ","End":"05:33.380","Text":"be a temperature which allows plus or minus."},{"Start":"05:33.380 ","End":"05:36.230","Text":"But if it\u0027s a temperature in centigrade,"},{"Start":"05:36.230 ","End":"05:39.500","Text":"it has to be bigger than minus 273,"},{"Start":"05:39.500 ","End":"05:41.795","Text":"for example, which absolute 0."},{"Start":"05:41.795 ","End":"05:46.190","Text":"In other words, when you have a problem which represents a situation in the real-world,"},{"Start":"05:46.190 ","End":"05:48.500","Text":"you have not only the mathematical limitation,"},{"Start":"05:48.500 ","End":"05:50.930","Text":"but you have the real-world limitation which"},{"Start":"05:50.930 ","End":"05:54.630","Text":"determines the domain of definition of a function."}],"ID":32355},{"Watched":false,"Name":"Graphical Description of a Function","Duration":"14m 52s","ChapterTopicVideoID":1101,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this clip, we\u0027ll talk about the graphical description of a function,"},{"Start":"00:03.690 ","End":"00:07.710","Text":"which is some kind of visual way of portraying,"},{"Start":"00:07.710 ","End":"00:09.240","Text":"presenting a function,"},{"Start":"00:09.240 ","End":"00:11.385","Text":"often very useful for understanding."},{"Start":"00:11.385 ","End":"00:13.365","Text":"People like visual things."},{"Start":"00:13.365 ","End":"00:16.499","Text":"Before we talk about drawing the graph of a function,"},{"Start":"00:16.499 ","End":"00:21.330","Text":"we need to go back to basics and talk about coordinate axes."},{"Start":"00:21.330 ","End":"00:23.355","Text":"What this involves is,"},{"Start":"00:23.355 ","End":"00:25.140","Text":"used to be on paper,"},{"Start":"00:25.140 ","End":"00:27.255","Text":"but now, it could be anywhere,"},{"Start":"00:27.255 ","End":"00:28.905","Text":"on my computer screen."},{"Start":"00:28.905 ","End":"00:30.810","Text":"There was a plane,"},{"Start":"00:30.810 ","End":"00:34.670","Text":"in the plane, there are 2 lines at right angles,"},{"Start":"00:34.670 ","End":"00:37.850","Text":"infinite, but I only have finite space here."},{"Start":"00:37.850 ","End":"00:45.560","Text":"The vertical one is known as the y-axis and the horizontal one is called the x-axis."},{"Start":"00:45.560 ","End":"00:49.215","Text":"The y-axis has a positive direction, which is upwards."},{"Start":"00:49.215 ","End":"00:51.410","Text":"The x-axis has a positive direction,"},{"Start":"00:51.410 ","End":"00:53.240","Text":"which is to the right."},{"Start":"00:53.240 ","End":"00:56.030","Text":"This part of it it\u0027s called the positive x-axis."},{"Start":"00:56.030 ","End":"01:01.250","Text":"This part of this is the positive y-axis and where they meet at the crosshairs,"},{"Start":"01:01.250 ","End":"01:03.020","Text":"that\u0027s called the origin,"},{"Start":"01:03.020 ","End":"01:05.795","Text":"and often denoted by the letter O."},{"Start":"01:05.795 ","End":"01:07.430","Text":"To put numbers on here,"},{"Start":"01:07.430 ","End":"01:10.055","Text":"we first of all need a scale."},{"Start":"01:10.055 ","End":"01:17.060","Text":"What we do is we divide it up using what are called tick marks like this."},{"Start":"01:17.060 ","End":"01:21.335","Text":"It doesn\u0027t have to be that each of this is one unit,"},{"Start":"01:21.335 ","End":"01:26.225","Text":"though it\u0027s commonly so that this might represent where x is 1,"},{"Start":"01:26.225 ","End":"01:29.330","Text":"2, 3, 4, and so on."},{"Start":"01:29.330 ","End":"01:32.315","Text":"But this is not necessary."},{"Start":"01:32.315 ","End":"01:36.500","Text":"Let me get my eraser and erase these."},{"Start":"01:36.500 ","End":"01:38.390","Text":"It could be something else."},{"Start":"01:38.390 ","End":"01:41.165","Text":"It could be that we draw every 5,"},{"Start":"01:41.165 ","End":"01:46.925","Text":"maybe this is 5, 10, 15, 20, etc."},{"Start":"01:46.925 ","End":"01:50.205","Text":"In which case, this would be minus 5,"},{"Start":"01:50.205 ","End":"01:52.660","Text":"minus 10, etc."},{"Start":"01:52.660 ","End":"01:54.740","Text":"As long as they\u0027re equal, it could be 10, 20,"},{"Start":"01:54.740 ","End":"01:56.120","Text":"30; 2, 4,"},{"Start":"01:56.120 ","End":"01:57.335","Text":"6, and so on."},{"Start":"01:57.335 ","End":"02:00.830","Text":"There\u0027s also a vertical scale with tick marks."},{"Start":"02:00.830 ","End":"02:02.180","Text":"When we draw them,"},{"Start":"02:02.180 ","End":"02:04.700","Text":"I don\u0027t even know if they should be the same,"},{"Start":"02:04.700 ","End":"02:07.355","Text":"drawn the same or not,"},{"Start":"02:07.355 ","End":"02:10.220","Text":"I don\u0027t think they have to be equally spaced."},{"Start":"02:10.220 ","End":"02:14.180","Text":"Certainly, the scale doesn\u0027t have to be the same as the scale."},{"Start":"02:14.180 ","End":"02:17.390","Text":"In other words, this could be 5, 10, 15,"},{"Start":"02:17.390 ","End":"02:19.280","Text":"whereas this might be 2,"},{"Start":"02:19.280 ","End":"02:22.325","Text":"4, 6, 8."},{"Start":"02:22.325 ","End":"02:25.220","Text":"Here too, you get negative on the other side,"},{"Start":"02:25.220 ","End":"02:26.705","Text":"this is 0 always."},{"Start":"02:26.705 ","End":"02:28.505","Text":"It could be minus 2,"},{"Start":"02:28.505 ","End":"02:30.830","Text":"minus 4, minus 6,"},{"Start":"02:30.830 ","End":"02:34.130","Text":"or it could be something completely different, 100, 200,"},{"Start":"02:34.130 ","End":"02:36.410","Text":"300, so often taken to be the same,"},{"Start":"02:36.410 ","End":"02:38.210","Text":"but very often also not."},{"Start":"02:38.210 ","End":"02:40.350","Text":"But the consistency is what\u0027s important."},{"Start":"02:40.350 ","End":"02:42.694","Text":"They should be spaced equally,"},{"Start":"02:42.694 ","End":"02:45.380","Text":"not just graphically, I mean the numbers as well."},{"Start":"02:45.380 ","End":"02:46.940","Text":"It can\u0027t have 5, 10,"},{"Start":"02:46.940 ","End":"02:48.815","Text":"and then 20, 30, 40."},{"Start":"02:48.815 ","End":"02:50.210","Text":"These are the axes,"},{"Start":"02:50.210 ","End":"02:52.465","Text":"yes, with the scale."},{"Start":"02:52.465 ","End":"02:58.220","Text":"What they enable us to do is to give any point in the plane,"},{"Start":"02:58.220 ","End":"02:59.435","Text":"a pair of numbers."},{"Start":"02:59.435 ","End":"03:03.395","Text":"Conversely, every repair of numbers is a point in the plane."},{"Start":"03:03.395 ","End":"03:06.160","Text":"Let me go back to the simple 1, 2, 3."},{"Start":"03:06.160 ","End":"03:08.240","Text":"Yeah, just to make it a bit simpler,"},{"Start":"03:08.240 ","End":"03:11.510","Text":"we\u0027ll make each one a single unit, 1, 2, 3, 4,"},{"Start":"03:11.510 ","End":"03:14.180","Text":"5, minus 1, minus 2, 1, 1,"},{"Start":"03:14.180 ","End":"03:17.180","Text":"2, 3, 4 minus 1 minus 2 for y."},{"Start":"03:17.180 ","End":"03:19.070","Text":"Let me show you an example."},{"Start":"03:19.070 ","End":"03:20.875","Text":"If I say to you,"},{"Start":"03:20.875 ","End":"03:23.170","Text":"the point 3, 2,"},{"Start":"03:23.170 ","End":"03:26.565","Text":"it\u0027s agreed that the first number,"},{"Start":"03:26.565 ","End":"03:33.189","Text":"the one on the left represents x and the other one represents the y."},{"Start":"03:33.189 ","End":"03:37.830","Text":"That means 3 to the east and 2 to the north,"},{"Start":"03:37.830 ","End":"03:43.160","Text":"if you like, or 3 along the positive x-axis and 2 along the y-axis."},{"Start":"03:43.160 ","End":"03:44.750","Text":"What you do in practice is,"},{"Start":"03:44.750 ","End":"03:48.905","Text":"I locate the 3 and I locate the 2 here."},{"Start":"03:48.905 ","End":"03:52.515","Text":"With dotted lines, I locate this point."},{"Start":"03:52.515 ","End":"03:56.605","Text":"This is the point which I call 3, 2."},{"Start":"03:56.605 ","End":"04:01.550","Text":"Conversely, if I\u0027m given a point in the plane,"},{"Start":"04:01.550 ","End":"04:08.480","Text":"for example, let\u0027s say this one here and you ask what are the coordinates of this point?"},{"Start":"04:08.480 ","End":"04:14.330","Text":"Then we drop perpendiculars to the y-axis and the x-axis."},{"Start":"04:14.330 ","End":"04:19.430","Text":"We see that where it hits the x-axis is the point minus"},{"Start":"04:19.430 ","End":"04:26.005","Text":"2 and x goes first and the y-axis at the point minus 1."},{"Start":"04:26.005 ","End":"04:29.690","Text":"This point is labeled minus 2 minus 1."},{"Start":"04:29.690 ","End":"04:30.920","Text":"It\u0027s a two-way thing."},{"Start":"04:30.920 ","End":"04:33.665","Text":"Each pair of numbers x and y,"},{"Start":"04:33.665 ","End":"04:35.000","Text":"gives me a point on the plane,"},{"Start":"04:35.000 ","End":"04:37.970","Text":"and each point on the plane gives me a pair of numbers, x, y."},{"Start":"04:37.970 ","End":"04:42.405","Text":"Here\u0027s another example, this point that\u0027s on the axis here,"},{"Start":"04:42.405 ","End":"04:46.485","Text":"basically is how many across the x-axis and how many up?"},{"Start":"04:46.485 ","End":"04:51.800","Text":"It\u0027s 4 in the x-direction and nothing up in the y direction,"},{"Start":"04:51.800 ","End":"04:54.725","Text":"or if you like, the perpendicular is just here at 0."},{"Start":"04:54.725 ","End":"04:58.055","Text":"This is the point 4, 0."},{"Start":"04:58.055 ","End":"05:03.930","Text":"Now, let\u0027s take a point on the y-axis, let\u0027s this one."},{"Start":"05:03.930 ","End":"05:09.860","Text":"This will be on the corresponding bit on the x-axis is 0."},{"Start":"05:09.860 ","End":"05:12.755","Text":"On the y-axis, we\u0027re up to minus 3."},{"Start":"05:12.755 ","End":"05:17.150","Text":"This point here is going to be 0 first,"},{"Start":"05:17.150 ","End":"05:21.155","Text":"in the x-direction and minus 3 in the y direction,"},{"Start":"05:21.155 ","End":"05:23.479","Text":"which means that it\u0027s not up, it\u0027s down."},{"Start":"05:23.479 ","End":"05:26.180","Text":"Of course, this point here is exceptional."},{"Start":"05:26.180 ","End":"05:29.600","Text":"At this point, which is the origin,"},{"Start":"05:29.600 ","End":"05:35.074","Text":"is 0, 0, that\u0027s the origin."},{"Start":"05:35.074 ","End":"05:39.530","Text":"Now that we have the relation between points and pairs of numbers,"},{"Start":"05:39.530 ","End":"05:43.850","Text":"now we can really start to draw graphs where typically we use y and"},{"Start":"05:43.850 ","End":"05:48.335","Text":"x and y is a function of x and that enables us to draw the graph."},{"Start":"05:48.335 ","End":"05:50.540","Text":"Before that, there\u0027s something I forgot to mention."},{"Start":"05:50.540 ","End":"05:53.780","Text":"Here\u0027s the place, the two lines,"},{"Start":"05:53.780 ","End":"05:58.610","Text":"the axes divide the plane into 4 parts,"},{"Start":"05:58.610 ","End":"06:01.055","Text":"into quarters, if you like."},{"Start":"06:01.055 ","End":"06:03.380","Text":"Each one is called a quadrant,"},{"Start":"06:03.380 ","End":"06:06.845","Text":"and this part here is called the first quadrant."},{"Start":"06:06.845 ","End":"06:10.670","Text":"Sometimes we write it as the first quadrant,"},{"Start":"06:10.670 ","End":"06:12.500","Text":"but in my days,"},{"Start":"06:12.500 ","End":"06:17.059","Text":"they used their Roman numerals, so in case you see that, this is the first quadrant,"},{"Start":"06:17.059 ","End":"06:19.415","Text":"this is the second quadrant."},{"Start":"06:19.415 ","End":"06:22.480","Text":"This is the third quadrant,"},{"Start":"06:22.480 ","End":"06:25.234","Text":"and this is the fourth quadrant."},{"Start":"06:25.234 ","End":"06:27.680","Text":"I\u0027ll take the example function,"},{"Start":"06:27.680 ","End":"06:31.370","Text":"one that we want to sketch as y equals x squared."},{"Start":"06:31.370 ","End":"06:33.830","Text":"Something very basic."},{"Start":"06:33.830 ","End":"06:37.400","Text":"What we do is partly it\u0027s an art,"},{"Start":"06:37.400 ","End":"06:39.530","Text":"but mostly, it\u0027s a precise thing."},{"Start":"06:39.530 ","End":"06:42.499","Text":"Usually, the art is getting the right scale."},{"Start":"06:42.499 ","End":"06:47.075","Text":"But I\u0027m also guessing how many numbers are to take in the sample."},{"Start":"06:47.075 ","End":"06:50.180","Text":"We basically make a table of values."},{"Start":"06:50.180 ","End":"06:54.605","Text":"Most of you probably at least vaguely remember doing this sort of thing."},{"Start":"06:54.605 ","End":"06:56.285","Text":"On the left column,"},{"Start":"06:56.285 ","End":"06:58.380","Text":"we call it x,"},{"Start":"06:58.380 ","End":"07:00.750","Text":"let me put values of x."},{"Start":"07:00.750 ","End":"07:05.465","Text":"Then we compute since y is the dependent variable,"},{"Start":"07:05.465 ","End":"07:06.650","Text":"y is the function of x."},{"Start":"07:06.650 ","End":"07:14.290","Text":"We first put in a value of x and then we find its corresponding unique y."},{"Start":"07:14.290 ","End":"07:17.140","Text":"Let\u0027s say x is 0."},{"Start":"07:17.140 ","End":"07:19.015","Text":"We\u0027ll take x equals 1,"},{"Start":"07:19.015 ","End":"07:22.285","Text":"x equals 2, maybe even 3."},{"Start":"07:22.285 ","End":"07:25.780","Text":"Don\u0027t have to use it in the end and let me compute y."},{"Start":"07:25.780 ","End":"07:27.460","Text":"Well, y is x squared,"},{"Start":"07:27.460 ","End":"07:30.715","Text":"so 0 squared is 0,"},{"Start":"07:30.715 ","End":"07:33.385","Text":"1 squared is 1,"},{"Start":"07:33.385 ","End":"07:35.380","Text":"2 squared is 4,"},{"Start":"07:35.380 ","End":"07:37.765","Text":"and 3 squared is 9."},{"Start":"07:37.765 ","End":"07:40.480","Text":"I don\u0027t think we\u0027ll fit very much more on this."},{"Start":"07:40.480 ","End":"07:41.995","Text":"Let me first see."},{"Start":"07:41.995 ","End":"07:44.484","Text":"Let\u0027s just draw something."},{"Start":"07:44.484 ","End":"07:47.560","Text":"I\u0027ll just us put it on pause while I draw."},{"Start":"07:47.560 ","End":"07:53.500","Text":"Here I am with my coordinate axes and I\u0027ve got some values."},{"Start":"07:53.500 ","End":"07:56.995","Text":"I want to decide on a scale that roughly,"},{"Start":"07:56.995 ","End":"07:59.365","Text":"if it\u0027s going to fit in."},{"Start":"07:59.365 ","End":"08:03.100","Text":"What I notice is that if I just need these,"},{"Start":"08:03.100 ","End":"08:08.785","Text":"all I need to go is up to 9 but I see that the y values generally become larger than x."},{"Start":"08:08.785 ","End":"08:12.520","Text":"What I\u0027m going to say that each unit here,"},{"Start":"08:12.520 ","End":"08:17.185","Text":"here there\u0027s going to be 1 unit, 1, 2, 3,"},{"Start":"08:17.185 ","End":"08:22.705","Text":"maybe 4 and here let\u0027s say that each tick is 2 units."},{"Start":"08:22.705 ","End":"08:26.510","Text":"They look the same size."},{"Start":"08:28.610 ","End":"08:32.010","Text":"If this is 1, 2, 3, 4,"},{"Start":"08:32.010 ","End":"08:35.475","Text":"and just to be different,"},{"Start":"08:35.475 ","End":"08:37.815","Text":"here, we\u0027ll have every 2; 2,"},{"Start":"08:37.815 ","End":"08:41.530","Text":"4, 6, 8, 10."},{"Start":"08:41.530 ","End":"08:43.960","Text":"Well, it\u0027s not just to be different because I saw I\u0027m going to have to fit"},{"Start":"08:43.960 ","End":"08:46.540","Text":"9 in and if I do it the same scale as this,"},{"Start":"08:46.540 ","End":"08:49.660","Text":"I won\u0027t get 9 in so I doubled up."},{"Start":"08:49.660 ","End":"08:54.505","Text":"Again, across here, it\u0027s going to be minus 1."},{"Start":"08:54.505 ","End":"08:57.775","Text":"See, as I say, are supposed to be equal, it\u0027s just me."},{"Start":"08:57.775 ","End":"09:00.880","Text":"Minus 3, minus 4,"},{"Start":"09:00.880 ","End":"09:04.310","Text":"and here, I can have,"},{"Start":"09:05.490 ","End":"09:08.980","Text":"though I won\u0027t really be needing these 3 as you\u0027ll see,"},{"Start":"09:08.980 ","End":"09:12.400","Text":"because I\u0027ll be squaring numbers that only come up positive,"},{"Start":"09:12.400 ","End":"09:14.170","Text":"but I didn\u0027t know that when I started."},{"Start":"09:14.170 ","End":"09:17.590","Text":"I\u0027m going to put in some minus 2,"},{"Start":"09:17.590 ","End":"09:22.370","Text":"minus 4, minus 6, and so on."},{"Start":"09:25.710 ","End":"09:28.555","Text":"I didn\u0027t put any negatives in here."},{"Start":"09:28.555 ","End":"09:34.270","Text":"Actually, that\u0027s going to be easy to do because if 0 squared is 0,"},{"Start":"09:34.270 ","End":"09:36.460","Text":"well, 0 is not a positive nor negative."},{"Start":"09:36.460 ","End":"09:41.695","Text":"But if 1 squared is 1 and minus 1 squared is also going to be 1,"},{"Start":"09:41.695 ","End":"09:50.160","Text":"minus 2 squared is also going to be 4 and minus 3 squared is also going to be 9."},{"Start":"09:50.160 ","End":"09:54.765","Text":"As you\u0027ll see, this will give me a certain symmetry."},{"Start":"09:54.765 ","End":"10:00.760","Text":"In fact, we can see it already if we plot 0, 0."},{"Start":"10:00.760 ","End":"10:04.780","Text":"I forgot to mention. From each of these pairs of x and y,"},{"Start":"10:04.780 ","End":"10:07.555","Text":"we get the coordinates of a point."},{"Start":"10:07.555 ","End":"10:09.355","Text":"This would be 0, 0,"},{"Start":"10:09.355 ","End":"10:10.750","Text":"this will be 1,"},{"Start":"10:10.750 ","End":"10:16.435","Text":"1, this 2, 4, 3, 9."},{"Start":"10:16.435 ","End":"10:19.585","Text":"The numbers don\u0027t have to be in any particular order in the table."},{"Start":"10:19.585 ","End":"10:23.665","Text":"Each one represents a point and we plot it whatever order we choose."},{"Start":"10:23.665 ","End":"10:27.100","Text":"We have the point minus 1, 1."},{"Start":"10:27.100 ","End":"10:29.660","Text":"That\u0027s a plus 1."},{"Start":"10:36.780 ","End":"10:39.830","Text":"Just a second."},{"Start":"10:40.980 ","End":"10:44.365","Text":"My pen was not writing."},{"Start":"10:44.365 ","End":"10:49.390","Text":"Minus 2, 4, minus 3,"},{"Start":"10:49.390 ","End":"10:55.420","Text":"9, these are the coordinates of the points and then we plot them as we saw earlier."},{"Start":"10:55.420 ","End":"10:57.670","Text":"I\u0027m going to use a different color,"},{"Start":"10:57.670 ","End":"11:02.740","Text":"it might help. Let\u0027s take blue."},{"Start":"11:02.740 ","End":"11:06.730","Text":"0, 0 would be here,"},{"Start":"11:06.730 ","End":"11:11.770","Text":"1, 1 be just 1, up here 1."},{"Start":"11:11.770 ","End":"11:13.540","Text":"Now, you\u0027ve seen this all before."},{"Start":"11:13.540 ","End":"11:16.510","Text":"I\u0027m not going to go into too much detail."},{"Start":"11:16.510 ","End":"11:23.390","Text":"2, 4 would be somewhere up here, somewhere across here."},{"Start":"11:23.460 ","End":"11:26.425","Text":"I could put the dotted lines in."},{"Start":"11:26.425 ","End":"11:28.330","Text":"I\u0027ll only have to erase them later."},{"Start":"11:28.330 ","End":"11:30.565","Text":"Maybe for one point I\u0027ll do it."},{"Start":"11:30.565 ","End":"11:36.760","Text":"2, 4 so that\u0027s this point and 3,"},{"Start":"11:36.760 ","End":"11:40.030","Text":"9 so 3 is here, 9 is here."},{"Start":"11:40.030 ","End":"11:42.695","Text":"Imagining the dotted lines."},{"Start":"11:42.695 ","End":"11:49.465","Text":"3, 9 be somewhere here then minus 1, 1."},{"Start":"11:49.465 ","End":"11:54.040","Text":"You can see what I was talking about symmetry before,"},{"Start":"11:54.040 ","End":"11:57.505","Text":"and then minus 2, 4,"},{"Start":"11:57.505 ","End":"12:01.525","Text":"the symmetry of this one."},{"Start":"12:01.525 ","End":"12:04.075","Text":"But because of my crooked drawing,"},{"Start":"12:04.075 ","End":"12:10.100","Text":"doesn\u0027t look so symmetrical and minus 3, 9."},{"Start":"12:12.780 ","End":"12:20.455","Text":"Then we tried to plot a line through these points,"},{"Start":"12:20.455 ","End":"12:22.510","Text":"but it\u0027ll come out okay,"},{"Start":"12:22.510 ","End":"12:25.700","Text":"but if not, not to worry."},{"Start":"12:26.220 ","End":"12:29.260","Text":"It\u0027s a bit too pointed and a bit lopsided,"},{"Start":"12:29.260 ","End":"12:30.400","Text":"but you get the idea."},{"Start":"12:30.400 ","End":"12:32.870","Text":"The idea certainly there."},{"Start":"12:33.000 ","End":"12:40.375","Text":"This is not a very good system altogether of figuring out what the curve looks like,"},{"Start":"12:40.375 ","End":"12:42.925","Text":"the method of plotting points."},{"Start":"12:42.925 ","End":"12:47.515","Text":"How do you know how to draw the line through the points?"},{"Start":"12:47.515 ","End":"12:52.510","Text":"Isn\u0027t it possible that, for example,"},{"Start":"12:52.510 ","End":"12:57.745","Text":"it was something completely different that went through these."},{"Start":"12:57.745 ","End":"12:59.560","Text":"Maybe this point is here,"},{"Start":"12:59.560 ","End":"13:08.335","Text":"but then it goes down here and then up here or here."},{"Start":"13:08.335 ","End":"13:13.060","Text":"The couple of zigzags through here and down to here."},{"Start":"13:13.060 ","End":"13:15.730","Text":"We don\u0027t really know which one is in faint,"},{"Start":"13:15.730 ","End":"13:18.790","Text":"but you get the idea just because you have a certain number of points,"},{"Start":"13:18.790 ","End":"13:21.430","Text":"you don\u0027t know how it behaves in between."},{"Start":"13:21.430 ","End":"13:23.470","Text":"Even if you draw a large numbers,"},{"Start":"13:23.470 ","End":"13:25.870","Text":"maybe the computer does it that way,"},{"Start":"13:25.870 ","End":"13:33.590","Text":"just lots and lots of little close to each other points and then draw it as best you can."},{"Start":"13:34.650 ","End":"13:42.520","Text":"Later learn how to explore and analyze functions from the equations"},{"Start":"13:42.520 ","End":"13:45.820","Text":"and we\u0027ll know that it has to have a minimum here and"},{"Start":"13:45.820 ","End":"13:49.600","Text":"curve convexity here and decrease here and so on,"},{"Start":"13:49.600 ","End":"13:51.160","Text":"all these kinds of things."},{"Start":"13:51.160 ","End":"13:58.330","Text":"But this straightforward point plotting is one way and"},{"Start":"13:58.330 ","End":"14:05.935","Text":"it\u0027s a quick way and usually it\u0027s roughly right but in principle it\u0027s not very founded."},{"Start":"14:05.935 ","End":"14:09.680","Text":"You really don\u0027t know what happens between the points."},{"Start":"14:11.580 ","End":"14:17.605","Text":"This function here, remember it\u0027s y equals x squared and also we should label the graph."},{"Start":"14:17.605 ","End":"14:21.865","Text":"I happen to be familiar with it so I know it does look something like this."},{"Start":"14:21.865 ","End":"14:24.490","Text":"But we do need some better technique and we will"},{"Start":"14:24.490 ","End":"14:27.800","Text":"learn them for finding out what a function looks like other"},{"Start":"14:27.800 ","End":"14:32.165","Text":"than lots and lots of points which you compute"},{"Start":"14:32.165 ","End":"14:40.170","Text":"and plot and then interpolate or put a line through."},{"Start":"14:40.170 ","End":"14:42.835","Text":"But it\u0027s good for rough and ready,"},{"Start":"14:42.835 ","End":"14:46.085","Text":"for quick idea, it almost always works."},{"Start":"14:46.085 ","End":"14:48.275","Text":"It gives you a rough idea."},{"Start":"14:48.275 ","End":"14:53.370","Text":"That\u0027s all I have to say at this point. That\u0027s all."}],"ID":1101},{"Watched":false,"Name":"Increase and Decrease of a Function","Duration":"6m 41s","ChapterTopicVideoID":8820,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this clip, we\u0027ll learn about the increase and decrease of a function."},{"Start":"00:03.945 ","End":"00:08.895","Text":"Here I\u0027ve already prepared a coordinate axis in a towel sketch."},{"Start":"00:08.895 ","End":"00:13.605","Text":"A function, say this is y equals f of x,"},{"Start":"00:13.605 ","End":"00:16.080","Text":"and in this pair of coordinate axis,"},{"Start":"00:16.080 ","End":"00:18.240","Text":"I\u0027ll draw something that looks a bit different,"},{"Start":"00:18.240 ","End":"00:21.600","Text":"let\u0027s say, this might also call it f of x."},{"Start":"00:21.600 ","End":"00:26.565","Text":"Now there\u0027s a difference in these 2 graphs and I\u0027d like to demonstrate it."},{"Start":"00:26.565 ","End":"00:29.895","Text":"If you noticed as you\u0027re going from left to right,"},{"Start":"00:29.895 ","End":"00:32.635","Text":"notice that you\u0027re climbing upwards."},{"Start":"00:32.635 ","End":"00:35.135","Text":"You\u0027re going along, and as you go along,"},{"Start":"00:35.135 ","End":"00:37.775","Text":"you\u0027re climbing up, like up a hill or something."},{"Start":"00:37.775 ","End":"00:39.350","Text":"Whereas in this 1,"},{"Start":"00:39.350 ","End":"00:41.570","Text":"as you\u0027re going from right to left,"},{"Start":"00:41.570 ","End":"00:45.889","Text":"you\u0027re actually going downhill or climbing down."},{"Start":"00:45.889 ","End":"00:49.040","Text":"This is called increasing and this is called decreasing."},{"Start":"00:49.040 ","End":"00:51.010","Text":"Now that\u0027s not very precise,"},{"Start":"00:51.010 ","End":"00:53.620","Text":"let\u0027s try and make it a little more formal."},{"Start":"00:53.620 ","End":"00:56.570","Text":"I\u0027ve taken off the arrows and we\u0027ll try and say"},{"Start":"00:56.570 ","End":"00:59.360","Text":"something more precise than going up a hill,"},{"Start":"00:59.360 ","End":"01:00.935","Text":"going down a hill, and so on."},{"Start":"01:00.935 ","End":"01:04.010","Text":"What I\u0027d like to do in the first case is take a couple of"},{"Start":"01:04.010 ","End":"01:07.865","Text":"points that say this point and this point."},{"Start":"01:07.865 ","End":"01:12.110","Text":"Each of these points has an x coordinate and the y coordinate,"},{"Start":"01:12.110 ","End":"01:13.970","Text":"let\u0027s say this is x_1,"},{"Start":"01:13.970 ","End":"01:17.720","Text":"y_1, and the other point will be x_2, y_2,"},{"Start":"01:17.720 ","End":"01:22.015","Text":"so here we have x_2 and here we have y_2."},{"Start":"01:22.015 ","End":"01:26.930","Text":"Now, notice that when we go from x_1 to x_2 it\u0027s in the positive direction."},{"Start":"01:26.930 ","End":"01:32.485","Text":"It\u0027s increasing as y goes from y_1 to y_2, it also increases."},{"Start":"01:32.485 ","End":"01:35.750","Text":"This function in this picture is increasing."},{"Start":"01:35.750 ","End":"01:40.895","Text":"What this means is besides the informal way of going up the hill and so forth,"},{"Start":"01:40.895 ","End":"01:46.065","Text":"is that if x_2 is bigger than x_1,"},{"Start":"01:46.065 ","End":"01:49.725","Text":"then y_2 is bigger than y_1."},{"Start":"01:49.725 ","End":"01:51.345","Text":"This is what I tried to show you here,"},{"Start":"01:51.345 ","End":"01:52.820","Text":"x_2 is bigger than x_1,"},{"Start":"01:52.820 ","End":"01:54.655","Text":"which means we\u0027re moving to the right and"},{"Start":"01:54.655 ","End":"01:56.910","Text":"y_2 is bigger than y_1, which means they\u0027re moving up."},{"Start":"01:56.910 ","End":"01:58.910","Text":"As we go right, we go upwards."},{"Start":"01:58.910 ","End":"02:03.680","Text":"Another way of saying this is that if x increases,"},{"Start":"02:03.680 ","End":"02:06.095","Text":"y increases, Let\u0027s take the other graph."},{"Start":"02:06.095 ","End":"02:10.340","Text":"In this picture, we have a picture of where f decreasing."},{"Start":"02:10.340 ","End":"02:15.485","Text":"That means again that if we take 2 points where this one is x_1,"},{"Start":"02:15.485 ","End":"02:18.875","Text":"y_1, and move to an adjacent point."},{"Start":"02:18.875 ","End":"02:20.300","Text":"Let\u0027s say this one,"},{"Start":"02:20.300 ","End":"02:22.510","Text":"which would be x_2,"},{"Start":"02:22.510 ","End":"02:25.305","Text":"y_2, we have a reversal here,"},{"Start":"02:25.305 ","End":"02:26.610","Text":"so f is decreasing,"},{"Start":"02:26.610 ","End":"02:31.740","Text":"which means that if x_2 is bigger than x_1,"},{"Start":"02:31.740 ","End":"02:34.980","Text":"and notice the reversal here,"},{"Start":"02:34.980 ","End":"02:38.785","Text":"y_2 was bigger than y_1 here y_2 is less than y_1."},{"Start":"02:38.785 ","End":"02:44.120","Text":"This formal thing says basically that if x increases,"},{"Start":"02:44.120 ","End":"02:48.230","Text":"then y decreases the opposite."},{"Start":"02:48.230 ","End":"02:51.570","Text":"The difference is that here if x_2 is bigger than x_1,"},{"Start":"02:51.570 ","End":"02:54.180","Text":"then y_2 is also bigger than y_1,"},{"Start":"02:54.180 ","End":"02:56.760","Text":"whereas here y_2 is less than y_1."},{"Start":"02:56.760 ","End":"02:58.880","Text":"In the slightly less formal here,"},{"Start":"02:58.880 ","End":"03:01.835","Text":"if x increases, then y increases."},{"Start":"03:01.835 ","End":"03:05.240","Text":"But in the other case, if x increases,"},{"Start":"03:05.240 ","End":"03:12.785","Text":"then y decreases, I could even say that y is f of x or just f even."},{"Start":"03:12.785 ","End":"03:17.150","Text":"That\u0027s why if f is called an increasing and decreasing because if x increases,"},{"Start":"03:17.150 ","End":"03:18.725","Text":"which is a natural way to go,"},{"Start":"03:18.725 ","End":"03:21.965","Text":"then here y increases and here y decreases."},{"Start":"03:21.965 ","End":"03:23.555","Text":"That\u0027s for this part."},{"Start":"03:23.555 ","End":"03:25.490","Text":"Next, we will be talking about how"},{"Start":"03:25.490 ","End":"03:29.765","Text":"single function can be also increasing and decreasing,"},{"Start":"03:29.765 ","End":"03:31.330","Text":"but in different parts."},{"Start":"03:31.330 ","End":"03:34.564","Text":"While you weren\u0027t looking I drew a little sketch."},{"Start":"03:34.564 ","End":"03:38.270","Text":"This is going to be an example of a graph or"},{"Start":"03:38.270 ","End":"03:43.055","Text":"a function which has both increasing and decreasing parts."},{"Start":"03:43.055 ","End":"03:46.085","Text":"I\u0027ll call this y equals f of x."},{"Start":"03:46.085 ","End":"03:47.944","Text":"Just looking at the picture,"},{"Start":"03:47.944 ","End":"03:54.605","Text":"you can see that the graph starts by going up as we\u0027re going all the time to the right."},{"Start":"03:54.605 ","End":"03:56.450","Text":"But as we go to the right here,"},{"Start":"03:56.450 ","End":"04:01.625","Text":"the y increases up to a certain point and it starts going down again,"},{"Start":"04:01.625 ","End":"04:04.295","Text":"and then we start going up again."},{"Start":"04:04.295 ","End":"04:10.100","Text":"Here we\u0027re increasing up to a certain point,"},{"Start":"04:10.100 ","End":"04:12.440","Text":"which happens to be 1,"},{"Start":"04:12.440 ","End":"04:16.340","Text":"that 1 itself we can\u0027t say it\u0027s neither increasing nor increasing,"},{"Start":"04:16.340 ","End":"04:17.840","Text":"but after the 1,"},{"Start":"04:17.840 ","End":"04:25.280","Text":"we start downwards till we get to where x equals 4, so there\u0027s decreasing."},{"Start":"04:25.280 ","End":"04:28.800","Text":"Then we start increasing again,"},{"Start":"04:29.210 ","End":"04:33.005","Text":"in general, this function continues indefinitely."},{"Start":"04:33.005 ","End":"04:35.570","Text":"Think I\u0027ll just emphasize it a bit with the arrows."},{"Start":"04:35.570 ","End":"04:39.440","Text":"Here we\u0027re going up and up and up,"},{"Start":"04:39.440 ","End":"04:42.544","Text":"and here we\u0027re going down and down,"},{"Start":"04:42.544 ","End":"04:46.430","Text":"and here we\u0027re going back up and up and so on."},{"Start":"04:46.430 ","End":"04:50.390","Text":"Now, a typical question might be as follows,"},{"Start":"04:50.390 ","End":"04:58.400","Text":"find the intervals of increase and decrease of the function,"},{"Start":"04:58.400 ","End":"05:00.740","Text":"I\u0027ll just say of f. First of all,"},{"Start":"05:00.740 ","End":"05:04.490","Text":"we\u0027ll look at the increase and we\u0027ll see that the increase is all the"},{"Start":"05:04.490 ","End":"05:08.390","Text":"way down from wherever some people would say minus infinity,"},{"Start":"05:08.390 ","End":"05:13.700","Text":"but it\u0027s just from to wherever up to the point where x equals 1."},{"Start":"05:13.700 ","End":"05:19.070","Text":"What concerns us, is actually not on the graph but on the x-axis,"},{"Start":"05:19.070 ","End":"05:20.330","Text":"so really what can I say,"},{"Start":"05:20.330 ","End":"05:27.230","Text":"the yellow, the shaded part up to here and here it\u0027s still increasing,"},{"Start":"05:27.230 ","End":"05:29.690","Text":"so I\u0027ll put some more lines like this."},{"Start":"05:29.690 ","End":"05:33.095","Text":"The decrease will be all the way from 1-4."},{"Start":"05:33.095 ","End":"05:41.465","Text":"What we would write as an answer is that the interval of increase is x is less than 1."},{"Start":"05:41.465 ","End":"05:46.115","Text":"Together with some people write or and some people write and I like to write,"},{"Start":"05:46.115 ","End":"05:52.220","Text":"or because each particular x can either be less than 1 or in this part here,"},{"Start":"05:52.220 ","End":"05:53.990","Text":"x is bigger than 4."},{"Start":"05:53.990 ","End":"05:58.125","Text":"For the decrease, the decrease is between 1 and 4,"},{"Start":"05:58.125 ","End":"06:01.234","Text":"so 1 is less than x,"},{"Start":"06:01.234 ","End":"06:02.570","Text":"is less than 4,"},{"Start":"06:02.570 ","End":"06:04.580","Text":"it means x is somewhere in here."},{"Start":"06:04.580 ","End":"06:07.460","Text":"Notice they use strict inequalities because"},{"Start":"06:07.460 ","End":"06:10.355","Text":"it\u0027s definitely increasing when x is less than 1."},{"Start":"06:10.355 ","End":"06:13.250","Text":"At 1 itself it\u0027s the border between"},{"Start":"06:13.250 ","End":"06:16.610","Text":"the increase and decrease and it\u0027s neither this nor that,"},{"Start":"06:16.610 ","End":"06:17.855","Text":"so it\u0027s not included."},{"Start":"06:17.855 ","End":"06:19.835","Text":"Similarly, with the 4,"},{"Start":"06:19.835 ","End":"06:22.240","Text":"it\u0027s between a decrease and an increase,"},{"Start":"06:22.240 ","End":"06:23.670","Text":"so I\u0027m not including it,"},{"Start":"06:23.670 ","End":"06:26.850","Text":"but between 1 and 4 were definitely decreasing."},{"Start":"06:26.850 ","End":"06:29.300","Text":"Then again, when x is bigger than 4,"},{"Start":"06:29.300 ","End":"06:32.420","Text":"this part here were definitely increasing again."},{"Start":"06:32.420 ","End":"06:34.850","Text":"Actually, there are 2 intervals of increase,"},{"Start":"06:34.850 ","End":"06:38.180","Text":"the 2 yellow bits and 1 interval of decrease."},{"Start":"06:38.180 ","End":"06:42.330","Text":"That\u0027s how we would write it and that answers the question."}],"ID":9010},{"Watched":false,"Name":"Positivity and Negativity of a Function Part 1","Duration":"4m 47s","ChapterTopicVideoID":9298,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.585","Text":"In this clip, we\u0027ll learn about the concept of positivity and negativity of a function,"},{"Start":"00:06.585 ","End":"00:10.320","Text":"and I\u0027ve brought with me a couple of coordinate axes to demonstrate."},{"Start":"00:10.320 ","End":"00:11.850","Text":"This 1 will be for the positivity,"},{"Start":"00:11.850 ","End":"00:13.650","Text":"this 1 for the negativity."},{"Start":"00:13.650 ","End":"00:17.415","Text":"Here I\u0027ll sketch something like the following."},{"Start":"00:17.415 ","End":"00:21.480","Text":"Well, if you look at any given point here on the graph,"},{"Start":"00:21.480 ","End":"00:23.490","Text":"the y is positive,"},{"Start":"00:23.490 ","End":"00:25.950","Text":"it\u0027s above the 0 line,"},{"Start":"00:25.950 ","End":"00:29.010","Text":"so this is positive, this is positive, positive, positive."},{"Start":"00:29.010 ","End":"00:32.280","Text":"Every x I choose will give me a positive y,"},{"Start":"00:32.280 ","End":"00:38.180","Text":"so that whenever I have that f of x is bigger than 0,"},{"Start":"00:38.180 ","End":"00:40.400","Text":"but not just for a particular x,"},{"Start":"00:40.400 ","End":"00:42.605","Text":"but for all x,"},{"Start":"00:42.605 ","End":"00:45.760","Text":"then f is called positive,"},{"Start":"00:45.760 ","End":"00:48.480","Text":"and I mean a positive function,"},{"Start":"00:48.480 ","End":"00:51.510","Text":"not that each individual x is positive."},{"Start":"00:51.510 ","End":"00:54.335","Text":"A number we know what it means for it to be positive,"},{"Start":"00:54.335 ","End":"00:56.255","Text":"but a function to be positive,"},{"Start":"00:56.255 ","End":"01:00.690","Text":"we say that when f is positive for all x."},{"Start":"01:00.690 ","End":"01:02.130","Text":"Now, graphically,"},{"Start":"01:02.130 ","End":"01:04.835","Text":"as an obvious graphical meaning."},{"Start":"01:04.835 ","End":"01:08.000","Text":"It means above the x-axis."},{"Start":"01:08.000 ","End":"01:12.170","Text":"See, the whole function is floating above the x-axis."},{"Start":"01:12.170 ","End":"01:14.780","Text":"It\u0027s always positive. Now, likewise,"},{"Start":"01:14.780 ","End":"01:16.730","Text":"you can then guess what I\u0027m going to do here."},{"Start":"01:16.730 ","End":"01:18.620","Text":"I\u0027m going to do something opposite."},{"Start":"01:18.620 ","End":"01:23.975","Text":"Notice that every given point here is negative, it\u0027s below 0."},{"Start":"01:23.975 ","End":"01:26.375","Text":"All of the points are negative."},{"Start":"01:26.375 ","End":"01:31.820","Text":"We have here f of x is strictly negative,"},{"Start":"01:31.820 ","End":"01:34.870","Text":"less than 0 for all x."},{"Start":"01:34.870 ","End":"01:36.255","Text":"Then we say,"},{"Start":"01:36.255 ","End":"01:37.680","Text":"under these circumstances,"},{"Start":"01:37.680 ","End":"01:39.390","Text":"that f is negative."},{"Start":"01:39.390 ","End":"01:43.535","Text":"We introduced 2 new concepts for a function,"},{"Start":"01:43.535 ","End":"01:49.255","Text":"and those concepts are positive and negative."},{"Start":"01:49.255 ","End":"01:55.500","Text":"In this case, it also has a graphical interpretation that when f is negative,"},{"Start":"01:55.500 ","End":"01:57.015","Text":"then, graphically,"},{"Start":"01:57.015 ","End":"02:01.405","Text":"the function f is below the x-axis."},{"Start":"02:01.405 ","End":"02:04.115","Text":"That\u0027s an easy way to spot it if you have the picture."},{"Start":"02:04.115 ","End":"02:06.090","Text":"Other than that, positive,"},{"Start":"02:06.090 ","End":"02:08.115","Text":"every x, when you substitute,"},{"Start":"02:08.115 ","End":"02:10.395","Text":"has to be bigger than 0,"},{"Start":"02:10.395 ","End":"02:12.105","Text":"and in the other case,"},{"Start":"02:12.105 ","End":"02:16.960","Text":"negative f of x has to be negative for whatever x you substitute."},{"Start":"02:16.960 ","End":"02:18.820","Text":"That\u0027s, I think, clear enough."},{"Start":"02:18.820 ","End":"02:21.740","Text":"The thing is not every function, in fact,"},{"Start":"02:21.740 ","End":"02:23.860","Text":"most functions are not positive or negative,"},{"Start":"02:23.860 ","End":"02:28.960","Text":"some functions have places where they\u0027re positive and places where they\u0027re negative."},{"Start":"02:28.960 ","End":"02:30.170","Text":"In the next slide,"},{"Start":"02:30.170 ","End":"02:35.270","Text":"we\u0027ll see that we have the mixture of positive and negative and what we do about it."},{"Start":"02:35.270 ","End":"02:39.530","Text":"Here, I\u0027m going to give you an example of the mixed variety,"},{"Start":"02:39.530 ","End":"02:41.315","Text":"which is most common."},{"Start":"02:41.315 ","End":"02:43.420","Text":"I\u0027ll draw a quick sketch."},{"Start":"02:43.420 ","End":"02:46.190","Text":"In this example, this is what we see,"},{"Start":"02:46.190 ","End":"02:48.110","Text":"and I\u0027ll also label the points."},{"Start":"02:48.110 ","End":"02:50.150","Text":"This is minus 4, let\u0027s say."},{"Start":"02:50.150 ","End":"02:52.865","Text":"This is 2. This looks like a 4."},{"Start":"02:52.865 ","End":"02:54.400","Text":"We have a function,"},{"Start":"02:54.400 ","End":"02:58.585","Text":"which is sometimes above the axis and sometimes below the axis,"},{"Start":"02:58.585 ","End":"03:01.160","Text":"and what I\u0027m going to do, first of all,"},{"Start":"03:01.160 ","End":"03:04.055","Text":"is to highlight just so we can see it visually."},{"Start":"03:04.055 ","End":"03:06.305","Text":"Let\u0027s go for the positive bits first."},{"Start":"03:06.305 ","End":"03:09.875","Text":"Positive means above the x-axis."},{"Start":"03:09.875 ","End":"03:15.915","Text":"This looks like we have positive here and here up to,"},{"Start":"03:15.915 ","End":"03:18.300","Text":"but not including the 0."},{"Start":"03:18.300 ","End":"03:20.870","Text":"Oh, look, it\u0027s above the axis here, too,"},{"Start":"03:20.870 ","End":"03:23.300","Text":"so we\u0027ll highlight this also,"},{"Start":"03:23.300 ","End":"03:26.360","Text":"and that highlights the positive bits."},{"Start":"03:26.360 ","End":"03:28.850","Text":"Negative means below the axis,"},{"Start":"03:28.850 ","End":"03:31.625","Text":"so we\u0027re negative over here,"},{"Start":"03:31.625 ","End":"03:35.490","Text":"and we\u0027re negative in this part here,"},{"Start":"03:35.490 ","End":"03:39.290","Text":"and the only other places remaining are the zeros themselves,"},{"Start":"03:39.290 ","End":"03:41.735","Text":"which are neither positive nor negative,"},{"Start":"03:41.735 ","End":"03:45.199","Text":"so we have negative 0, positive,"},{"Start":"03:45.199 ","End":"03:50.130","Text":"0, negative 0, and positive again."},{"Start":"03:50.130 ","End":"03:54.275","Text":"I would like to warn you not to confuse"},{"Start":"03:54.275 ","End":"03:59.920","Text":"positivity and negativity with increase and decrease."},{"Start":"03:59.920 ","End":"04:01.955","Text":"You have been warned,"},{"Start":"04:01.955 ","End":"04:05.570","Text":"but I will also go into it further as a good reminder."},{"Start":"04:05.570 ","End":"04:07.610","Text":"We start up here with the function."},{"Start":"04:07.610 ","End":"04:09.080","Text":"We\u0027re going from left to right."},{"Start":"04:09.080 ","End":"04:12.380","Text":"We start up from negative, but we\u0027re increasing."},{"Start":"04:12.380 ","End":"04:17.795","Text":"Now, we continue increasing here all the way up to this point here,"},{"Start":"04:17.795 ","End":"04:20.570","Text":"which has nothing to do with negativity and positivity,"},{"Start":"04:20.570 ","End":"04:21.800","Text":"but in the case of increase,"},{"Start":"04:21.800 ","End":"04:23.060","Text":"this is where the increase stops."},{"Start":"04:23.060 ","End":"04:24.200","Text":"The decrease starts,"},{"Start":"04:24.200 ","End":"04:25.595","Text":"where it was still positive,"},{"Start":"04:25.595 ","End":"04:28.610","Text":"and once we\u0027re past the 0, we\u0027re still decreasing,"},{"Start":"04:28.610 ","End":"04:29.960","Text":"but now we\u0027re negative,"},{"Start":"04:29.960 ","End":"04:31.915","Text":"and then we get to some point here,"},{"Start":"04:31.915 ","End":"04:33.410","Text":"and then we\u0027re still negative,"},{"Start":"04:33.410 ","End":"04:35.050","Text":"but we\u0027ll now reverse direction."},{"Start":"04:35.050 ","End":"04:38.910","Text":"We\u0027re increasing, and then continue past the 0,"},{"Start":"04:38.910 ","End":"04:40.440","Text":"and now we\u0027re still increasing,"},{"Start":"04:40.440 ","End":"04:41.700","Text":"but we\u0027re positive,"},{"Start":"04:41.700 ","End":"04:44.585","Text":"so I hope that clears up negativity,"},{"Start":"04:44.585 ","End":"04:48.450","Text":"positivity, and increase and decrease."}],"ID":9610},{"Watched":false,"Name":"Positivity and Negativity of a Function Part 2","Duration":"3m 12s","ChapterTopicVideoID":9299,"CourseChapterTopicPlaylistID":1171,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"I want to finish with a book question."},{"Start":"00:02.760 ","End":"00:08.130","Text":"A book question might ask you to do the following exercise question and this is it."},{"Start":"00:08.130 ","End":"00:14.505","Text":"Find intervals of positivity and negativity in the function f,"},{"Start":"00:14.505 ","End":"00:19.460","Text":"and so the answer will be the green bit will be the negative and the,"},{"Start":"00:19.460 ","End":"00:21.540","Text":"which is this orange will be the positive,"},{"Start":"00:21.540 ","End":"00:24.165","Text":"but not exactly in this form."},{"Start":"00:24.165 ","End":"00:29.400","Text":"I should have stressed that what we\u0027re concerned about is only the x values."},{"Start":"00:29.400 ","End":"00:34.200","Text":"Emphasize the orange bit on the x-axis,"},{"Start":"00:34.200 ","End":"00:37.795","Text":"and that will be this bit."},{"Start":"00:37.795 ","End":"00:41.790","Text":"Also from here onwards,"},{"Start":"00:41.790 ","End":"00:44.030","Text":"the above, these values of x,"},{"Start":"00:44.030 ","End":"00:46.835","Text":"it\u0027s positive and above these values of x."},{"Start":"00:46.835 ","End":"00:48.515","Text":"As for the other 1,"},{"Start":"00:48.515 ","End":"00:50.060","Text":"the green on the graph,"},{"Start":"00:50.060 ","End":"00:51.290","Text":"this is where it\u0027s negative,"},{"Start":"00:51.290 ","End":"00:55.970","Text":"but the x is for which f of x is negative here."},{"Start":"00:55.970 ","End":"00:57.855","Text":"For this f of x is negative,"},{"Start":"00:57.855 ","End":"01:01.100","Text":"and for this x for x is negative."},{"Start":"01:01.100 ","End":"01:06.620","Text":"The way we write it is just in terms of inequalities and x."},{"Start":"01:06.620 ","End":"01:09.500","Text":"We write the answer as, first of all,"},{"Start":"01:09.500 ","End":"01:11.300","Text":"I\u0027ll write the positivity,"},{"Start":"01:11.300 ","End":"01:13.885","Text":"and then I\u0027ll write the negativity."},{"Start":"01:13.885 ","End":"01:17.920","Text":"Positivity, the orange minus 4,"},{"Start":"01:17.920 ","End":"01:19.865","Text":"less than x, less than 2,"},{"Start":"01:19.865 ","End":"01:21.800","Text":"and this other bit here,"},{"Start":"01:21.800 ","End":"01:24.034","Text":"so we write or in between,"},{"Start":"01:24.034 ","End":"01:26.195","Text":"because x could be here or here,"},{"Start":"01:26.195 ","End":"01:28.745","Text":"x is greater than 4."},{"Start":"01:28.745 ","End":"01:30.815","Text":"Similarly, if you look at the green bits,"},{"Start":"01:30.815 ","End":"01:35.775","Text":"you\u0027ll see that this is x is less than minus 4,"},{"Start":"01:35.775 ","End":"01:37.560","Text":"or the other green bit,"},{"Start":"01:37.560 ","End":"01:42.090","Text":"2 less than x, less than 4."},{"Start":"01:42.090 ","End":"01:44.930","Text":"That\u0027s it for positivity and negativity,"},{"Start":"01:44.930 ","End":"01:49.280","Text":"but we\u0027re not done because there\u0027s 1 more concept I want to introduce you to,"},{"Start":"01:49.280 ","End":"01:54.265","Text":"and that is called non-negativity on the next slide."},{"Start":"01:54.265 ","End":"01:58.490","Text":"Here we are in the last slide we\u0027re going to talk about non-negativity."},{"Start":"01:58.490 ","End":"02:00.545","Text":"Let me just write that word."},{"Start":"02:00.545 ","End":"02:02.665","Text":"Here\u0027s the example."},{"Start":"02:02.665 ","End":"02:04.750","Text":"Now this almost looks like"},{"Start":"02:04.750 ","End":"02:08.650","Text":"a positive function with the exception as I\u0027ve been pointing out,"},{"Start":"02:08.650 ","End":"02:13.645","Text":"that it does actually touch the x-axis at a certain number of points."},{"Start":"02:13.645 ","End":"02:16.365","Text":"Where it touches it\u0027s not positive,"},{"Start":"02:16.365 ","End":"02:19.045","Text":"it\u0027s not negative either, it\u0027s 0."},{"Start":"02:19.045 ","End":"02:24.370","Text":"Sometimes it\u0027s useful to consider positive and 0 together and for that,"},{"Start":"02:24.370 ","End":"02:28.760","Text":"something is non-negative, then it means it can be positive or 0."},{"Start":"02:28.760 ","End":"02:31.585","Text":"For example, 7 is non-negative,"},{"Start":"02:31.585 ","End":"02:34.525","Text":"might be 7 and 0 is non-negative."},{"Start":"02:34.525 ","End":"02:38.290","Text":"Minus 5 isn\u0027t non-negative, it\u0027s downright negative,"},{"Start":"02:38.290 ","End":"02:42.675","Text":"but positive and negative together collected is non-negative."},{"Start":"02:42.675 ","End":"02:45.115","Text":"Here\u0027s an example of a function which is"},{"Start":"02:45.115 ","End":"02:49.070","Text":"non-negative and specifically I have to require that for each x,"},{"Start":"02:49.070 ","End":"02:52.115","Text":"f of x either be positive or 0."},{"Start":"02:52.115 ","End":"02:54.500","Text":"We write that f of x,"},{"Start":"02:54.500 ","End":"03:00.105","Text":"instead of writing bigger than 0 or bigger or equal to 0 for all x."},{"Start":"03:00.105 ","End":"03:04.820","Text":"Then f is called non-negative and"},{"Start":"03:04.820 ","End":"03:09.590","Text":"graphically it means it\u0027s above the axis or possibly touching."},{"Start":"03:09.590 ","End":"03:13.140","Text":"I\u0027m done for this presentation."}],"ID":9611}],"Thumbnail":null,"ID":1171},{"Name":"Common Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Linear Function","Duration":"3m 8s","ChapterTopicVideoID":8823,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this clip, we introduced the concept of a linear function."},{"Start":"00:04.020 ","End":"00:13.455","Text":"A linear function is a function of the form y or f of x is equal to mx plus n,"},{"Start":"00:13.455 ","End":"00:16.680","Text":"where m and n are just some numbers, some constants."},{"Start":"00:16.680 ","End":"00:18.120","Text":"Often in the literature,"},{"Start":"00:18.120 ","End":"00:22.800","Text":"1 sees it written as y equals ax plus b,"},{"Start":"00:22.800 ","End":"00:24.090","Text":"very common letters,"},{"Start":"00:24.090 ","End":"00:26.850","Text":"and I\u0027ve seen all sorts of hybrid combinations."},{"Start":"00:26.850 ","End":"00:31.875","Text":"I\u0027ve seen y equals mx plus b. I\u0027ll give you an example."},{"Start":"00:31.875 ","End":"00:37.935","Text":"A linear function could be y equals 4x plus 10."},{"Start":"00:37.935 ","End":"00:39.870","Text":"That\u0027s a specific example and it doesn\u0027t really"},{"Start":"00:39.870 ","End":"00:42.315","Text":"matter what letters we use in the general."},{"Start":"00:42.315 ","End":"00:46.390","Text":"In this case we would have that m is equal to 4,"},{"Start":"00:46.390 ","End":"00:48.400","Text":"and n is equal to 10."},{"Start":"00:48.400 ","End":"00:50.530","Text":"For any pair of numbers we give for m and n,"},{"Start":"00:50.530 ","End":"00:51.850","Text":"we get a linear function."},{"Start":"00:51.850 ","End":"00:58.805","Text":"Here\u0027s another example, y equals minus 10x. What do we get?"},{"Start":"00:58.805 ","End":"01:03.105","Text":"Here, m is equal to minus 10x."},{"Start":"01:03.105 ","End":"01:06.000","Text":"What about the n? Well, there is no n. So in other words,"},{"Start":"01:06.000 ","End":"01:07.695","Text":"n is equal to 0."},{"Start":"01:07.695 ","End":"01:11.815","Text":"Yet another example, y is equal to 8."},{"Start":"01:11.815 ","End":"01:14.320","Text":"Here we don\u0027t have anything with x."},{"Start":"01:14.320 ","End":"01:19.275","Text":"That means that M is 0 and in this case n is 8."},{"Start":"01:19.275 ","End":"01:23.310","Text":"See if I write minus 10x plus 0 as here,"},{"Start":"01:23.310 ","End":"01:24.750","Text":"and that\u0027s minus 10x."},{"Start":"01:24.750 ","End":"01:26.700","Text":"If I write 0x plus 8,"},{"Start":"01:26.700 ","End":"01:29.290","Text":"there\u0027s no need for the 0x, so it\u0027s just 8."},{"Start":"01:29.290 ","End":"01:35.524","Text":"In fact, you can get more extreme situations with almost nothing left in the equation."},{"Start":"01:35.524 ","End":"01:40.460","Text":"For example, y equals 0 is also a linear equation."},{"Start":"01:40.460 ","End":"01:43.849","Text":"It\u0027s just that m and n are both 0."},{"Start":"01:43.849 ","End":"01:46.310","Text":"Because if I put m and n both 0 here,"},{"Start":"01:46.310 ","End":"01:49.670","Text":"I just get y equals 0 and this too is a linear equation."},{"Start":"01:49.670 ","End":"01:53.930","Text":"Sometimes something appears to be a linear equation is in fact,"},{"Start":"01:53.930 ","End":"01:56.075","Text":"or can be brought into a linear form."},{"Start":"01:56.075 ","End":"02:02.660","Text":"For example, I could take the equation 4y plus 2x equals 10."},{"Start":"02:02.660 ","End":"02:05.225","Text":"It\u0027s an equation, but it\u0027s not a function."},{"Start":"02:05.225 ","End":"02:08.630","Text":"It turns out that you can actually make y as a linear function"},{"Start":"02:08.630 ","End":"02:12.120","Text":"of x just by doing a bit of algebra. Let\u0027s see."},{"Start":"02:12.120 ","End":"02:15.515","Text":"If we bring the 2x to the other side,"},{"Start":"02:15.515 ","End":"02:22.055","Text":"we will get 4y is equal to minus 2x plus 10."},{"Start":"02:22.055 ","End":"02:23.855","Text":"Then dividing by 4,"},{"Start":"02:23.855 ","End":"02:27.905","Text":"we get that y is equal to minus 2 over 4."},{"Start":"02:27.905 ","End":"02:29.470","Text":"Let\u0027s cancel the fraction."},{"Start":"02:29.470 ","End":"02:35.630","Text":"That will be minus 1/2 of x plus 10 over 4 is 5 over 2 or 2.5."},{"Start":"02:35.630 ","End":"02:38.040","Text":"I\u0027ll write it as 5 over 2."},{"Start":"02:38.040 ","End":"02:42.530","Text":"This equation actually represented y as a linear function of x."},{"Start":"02:42.530 ","End":"02:51.405","Text":"We can say that m is equal to minus 1/2 and that n is equal to 5 over 2."},{"Start":"02:51.405 ","End":"02:53.000","Text":"We haven\u0027t done very much so far."},{"Start":"02:53.000 ","End":"02:56.975","Text":"We\u0027ve just shown essentially what a linear function is,"},{"Start":"02:56.975 ","End":"02:58.340","Text":"and this is the form,"},{"Start":"02:58.340 ","End":"02:59.645","Text":"mx plus n,"},{"Start":"02:59.645 ","End":"03:02.045","Text":"general linear function in x."},{"Start":"03:02.045 ","End":"03:05.690","Text":"As I\u0027ve said, some other textbooks and so on using different letters,"},{"Start":"03:05.690 ","End":"03:07.280","Text":"but we don\u0027t let that worry us."},{"Start":"03:07.280 ","End":"03:09.480","Text":"Okay, that\u0027s it for now."}],"ID":9011},{"Watched":false,"Name":"The Quadratic Function","Duration":"2m 25s","ChapterTopicVideoID":8824,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"This clip is a brief introduction to the Quadratic Function."},{"Start":"00:04.050 ","End":"00:09.675","Text":"Quadratic function is a function of the form f of x is"},{"Start":"00:09.675 ","End":"00:15.900","Text":"equal to ax squared plus bx plus c,"},{"Start":"00:15.900 ","End":"00:18.600","Text":"where a, b, and c are some constants."},{"Start":"00:18.600 ","End":"00:23.460","Text":"It\u0027s very important that a cannot be 0 because if a is 0,"},{"Start":"00:23.460 ","End":"00:24.990","Text":"it\u0027s not a quadratic function."},{"Start":"00:24.990 ","End":"00:26.460","Text":"There\u0027s no x squared term,"},{"Start":"00:26.460 ","End":"00:27.765","Text":"it\u0027s a linear function."},{"Start":"00:27.765 ","End":"00:30.240","Text":"I\u0027ll give you an example to start off with."},{"Start":"00:30.240 ","End":"00:36.800","Text":"F of x is equal to 4x squared minus 10x plus 1."},{"Start":"00:36.800 ","End":"00:42.394","Text":"This is a quadratic function with the constants a being equal to 4,"},{"Start":"00:42.394 ","End":"00:45.050","Text":"b being minus 10,"},{"Start":"00:45.050 ","End":"00:47.030","Text":"and c being 1."},{"Start":"00:47.030 ","End":"00:50.600","Text":"Another example, and here I\u0027ll use the y notation."},{"Start":"00:50.600 ","End":"00:55.294","Text":"Y is equal to minus x squared plus 20."},{"Start":"00:55.294 ","End":"00:58.880","Text":"Here, a is equal to minus 1."},{"Start":"00:58.880 ","End":"01:00.875","Text":"That\u0027s the coefficient of x squared."},{"Start":"01:00.875 ","End":"01:03.200","Text":"B is absent. It\u0027s missing."},{"Start":"01:03.200 ","End":"01:07.060","Text":"When it\u0027s missing, it\u0027s 0 because you could say plus 0x here."},{"Start":"01:07.060 ","End":"01:09.170","Text":"That\u0027s for c, it\u0027s just 20."},{"Start":"01:09.170 ","End":"01:14.265","Text":"Another example, g of x is 50x squared minus 20x."},{"Start":"01:14.265 ","End":"01:16.550","Text":"Now here, in the previous example,"},{"Start":"01:16.550 ","End":"01:17.900","Text":"we had b equals 0."},{"Start":"01:17.900 ","End":"01:20.870","Text":"Here c is missing the 3 coefficient,"},{"Start":"01:20.870 ","End":"01:24.305","Text":"so we get a is equal to 50,"},{"Start":"01:24.305 ","End":"01:27.540","Text":"b would be minus 20 in this case,"},{"Start":"01:27.540 ","End":"01:29.375","Text":"and since there is no other term,"},{"Start":"01:29.375 ","End":"01:31.655","Text":"it\u0027s like saying that c equals 0."},{"Start":"01:31.655 ","End":"01:33.965","Text":"Let\u0027s take 1 more example."},{"Start":"01:33.965 ","End":"01:38.415","Text":"Y is equal to minus 4x squared,"},{"Start":"01:38.415 ","End":"01:42.195","Text":"just like here, b was 0 and here c was 0."},{"Start":"01:42.195 ","End":"01:46.130","Text":"Here, both b and c are 0. We\u0027re only left with a."},{"Start":"01:46.130 ","End":"01:49.249","Text":"A is minus 4, b is missing,"},{"Start":"01:49.249 ","End":"01:51.450","Text":"and the c term is missing,"},{"Start":"01:51.450 ","End":"01:53.195","Text":"which means that c is 0."},{"Start":"01:53.195 ","End":"01:55.630","Text":"That\u0027s as far as the examples go."},{"Start":"01:55.630 ","End":"01:59.345","Text":"In the rest of the tutorial on the quadratic function,"},{"Start":"01:59.345 ","End":"02:02.180","Text":"we\u0027ll learn things like what is the meaning of a, b,"},{"Start":"02:02.180 ","End":"02:05.194","Text":"and c, the graph of the quadratic function,"},{"Start":"02:05.194 ","End":"02:08.795","Text":"which happens to be in the shape of a parabola."},{"Start":"02:08.795 ","End":"02:09.860","Text":"I\u0027ll just write the word,"},{"Start":"02:09.860 ","End":"02:11.645","Text":"but we\u0027ll encounter it later."},{"Start":"02:11.645 ","End":"02:14.660","Text":"We\u0027ll learn about the vertex of the parabola,"},{"Start":"02:14.660 ","End":"02:19.189","Text":"we\u0027ll learn about the intersection of the quadratic function with the axes,"},{"Start":"02:19.189 ","End":"02:22.955","Text":"the x and y, and all sorts of other fun stuff like that."},{"Start":"02:22.955 ","End":"02:25.890","Text":"But we\u0027re done for the intro."}],"ID":9012},{"Watched":false,"Name":"The Exponential Function","Duration":"7m 16s","ChapterTopicVideoID":8821,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this section, we\u0027ll talk about the exponential function."},{"Start":"00:03.540 ","End":"00:09.915","Text":"The exponential function looks like f of x is equal to a^x,"},{"Start":"00:09.915 ","End":"00:12.600","Text":"where a is a constant or parameter,"},{"Start":"00:12.600 ","End":"00:14.280","Text":"but there is a restriction."},{"Start":"00:14.280 ","End":"00:18.615","Text":"The restriction is that a has to be bigger than 0,"},{"Start":"00:18.615 ","End":"00:21.480","Text":"and it also should not equal 1."},{"Start":"00:21.480 ","End":"00:25.010","Text":"In other words, it\u0027s a positive number not equal to 1."},{"Start":"00:25.010 ","End":"00:26.330","Text":"This is the function."},{"Start":"00:26.330 ","End":"00:32.080","Text":"The reason it\u0027s called exponential is that the variable x is the exponent."},{"Start":"00:32.080 ","End":"00:33.975","Text":"A also has a name, it\u0027s the base."},{"Start":"00:33.975 ","End":"00:35.835","Text":"We have base and exponent."},{"Start":"00:35.835 ","End":"00:38.240","Text":"In a lot of books and a lot of colleges,"},{"Start":"00:38.240 ","End":"00:42.630","Text":"they use b^x rather than a^x,"},{"Start":"00:42.630 ","End":"00:48.170","Text":"because then x is like exponent and b is base."},{"Start":"00:48.170 ","End":"00:49.850","Text":"But here we\u0027ll use the letter a."},{"Start":"00:49.850 ","End":"00:55.700","Text":"Some examples we could take f of x is equal to 4^x."},{"Start":"00:55.700 ","End":"00:58.535","Text":"Or we could take and say same,"},{"Start":"00:58.535 ","End":"00:59.900","Text":"use the y notation,"},{"Start":"00:59.900 ","End":"01:02.375","Text":"y equals 1/2^x,"},{"Start":"01:02.375 ","End":"01:07.100","Text":"or y equals 4.7^x,"},{"Start":"01:07.100 ","End":"01:08.855","Text":"something to the power of x."},{"Start":"01:08.855 ","End":"01:11.570","Text":"In case you\u0027re wondering about these restrictions about"},{"Start":"01:11.570 ","End":"01:14.645","Text":"why a has to be positive and not equal to 1."},{"Start":"01:14.645 ","End":"01:16.850","Text":"Just mention this if a equals 1,"},{"Start":"01:16.850 ","End":"01:19.100","Text":"then we just get the constant function."},{"Start":"01:19.100 ","End":"01:22.010","Text":"If a is 1, 1^x is always 1."},{"Start":"01:22.010 ","End":"01:24.815","Text":"A is 1 we get a constant function."},{"Start":"01:24.815 ","End":"01:26.555","Text":"If a is negative,"},{"Start":"01:26.555 ","End":"01:28.745","Text":"we have problems with the domain."},{"Start":"01:28.745 ","End":"01:32.765","Text":"For instance, if we take a is equal to minus 4,"},{"Start":"01:32.765 ","End":"01:35.420","Text":"then we have a problem because it isn\u0027t defined for all x."},{"Start":"01:35.420 ","End":"01:40.520","Text":"Say you put x equals 1/2 minus 4^1/2 square root of minus 4."},{"Start":"01:40.520 ","End":"01:41.885","Text":"There are various problems."},{"Start":"01:41.885 ","End":"01:44.540","Text":"In short, a^x is the exponential function,"},{"Start":"01:44.540 ","End":"01:46.940","Text":"but there are some restrictions that you have to observe."},{"Start":"01:46.940 ","End":"01:49.405","Text":"Fortunately with the exponential function,"},{"Start":"01:49.405 ","End":"01:52.105","Text":"there\u0027s only 1 parameter, that\u0027s a,"},{"Start":"01:52.105 ","End":"01:55.530","Text":"in contrast to say the quadratic function, which has an a, a b,"},{"Start":"01:55.530 ","End":"01:59.285","Text":"and a c. We only have to worry about the parameter a."},{"Start":"01:59.285 ","End":"02:02.325","Text":"There\u0027s really 2 main possibilities for a."},{"Start":"02:02.325 ","End":"02:03.904","Text":"Because if I look at this restriction,"},{"Start":"02:03.904 ","End":"02:06.740","Text":"I\u0027ll see that either a bigger than 1,"},{"Start":"02:06.740 ","End":"02:10.615","Text":"or a can be between 0 and 1."},{"Start":"02:10.615 ","End":"02:15.400","Text":"That distinguishes these 2 cases is that if a is bigger than 1,"},{"Start":"02:15.400 ","End":"02:17.860","Text":"then the function is increasing."},{"Start":"02:17.860 ","End":"02:20.365","Text":"If a is between 0 and 1,"},{"Start":"02:20.365 ","End":"02:22.705","Text":"then f is decreasing."},{"Start":"02:22.705 ","End":"02:25.685","Text":"I\u0027d like to illustrate this to you graphically."},{"Start":"02:25.685 ","End":"02:27.210","Text":"Here are the axes,"},{"Start":"02:27.210 ","End":"02:29.825","Text":"and I\u0027ll draw both of them on the same graph."},{"Start":"02:29.825 ","End":"02:31.930","Text":"I\u0027ll take an example of a bigger than 1,"},{"Start":"02:31.930 ","End":"02:34.525","Text":"let\u0027s say a equals 2."},{"Start":"02:34.525 ","End":"02:38.860","Text":"We\u0027re going to draw y equals 2^x."},{"Start":"02:38.860 ","End":"02:42.015","Text":"If x is 0, 2^0 is 1."},{"Start":"02:42.015 ","End":"02:44.685","Text":"This might be 0, 1."},{"Start":"02:44.685 ","End":"02:47.715","Text":"If x is 1, then y is 2."},{"Start":"02:47.715 ","End":"02:50.265","Text":"If x is 2,"},{"Start":"02:50.265 ","End":"02:52.560","Text":"then y equals 2^2,"},{"Start":"02:52.560 ","End":"02:55.465","Text":"which is 4, increases steeply."},{"Start":"02:55.465 ","End":"03:01.700","Text":"We can go on the other side and say x is equal to minus 1."},{"Start":"03:01.700 ","End":"03:04.985","Text":"2^minus 1 would be 1/2."},{"Start":"03:04.985 ","End":"03:06.935","Text":"If I take x as minus 2,"},{"Start":"03:06.935 ","End":"03:09.670","Text":"2^minus 2, we get 1/4."},{"Start":"03:09.670 ","End":"03:12.330","Text":"If I join a line through them,"},{"Start":"03:12.330 ","End":"03:15.410","Text":"that\u0027s the general idea of y equals 2^x."},{"Start":"03:15.410 ","End":"03:17.330","Text":"You notice that it\u0027s increasing."},{"Start":"03:17.330 ","End":"03:20.750","Text":"That\u0027s typical for a bigger than 1."},{"Start":"03:20.750 ","End":"03:23.300","Text":"Let\u0027s now do an example of a less than 1,"},{"Start":"03:23.300 ","End":"03:24.950","Text":"say a equals 1/2,"},{"Start":"03:24.950 ","End":"03:28.590","Text":"and I\u0027ll do that in a different color to the power of x."},{"Start":"03:28.590 ","End":"03:30.979","Text":"Again, we can sketch a few points."},{"Start":"03:30.979 ","End":"03:34.980","Text":"When x is 0, y is again equal to 1."},{"Start":"03:34.980 ","End":"03:41.690","Text":"In fact, that\u0027s true for all of the exponential functions that f of 0 is equal to 1."},{"Start":"03:41.690 ","End":"03:43.220","Text":"They all pass through this point."},{"Start":"03:43.220 ","End":"03:45.560","Text":"This time, when x is 1,"},{"Start":"03:45.560 ","End":"03:47.675","Text":"y is equal to 1/2."},{"Start":"03:47.675 ","End":"03:50.435","Text":"We get this point here,"},{"Start":"03:50.435 ","End":"03:52.955","Text":"which is actually the mirror image of this point."},{"Start":"03:52.955 ","End":"03:55.000","Text":"If I let equal 2,"},{"Start":"03:55.000 ","End":"03:56.960","Text":"then y equals 1/2^2,"},{"Start":"03:56.960 ","End":"03:58.400","Text":"which is 1/4,"},{"Start":"03:58.400 ","End":"04:02.330","Text":"which is exactly the mirror image of that one there."},{"Start":"04:02.330 ","End":"04:04.820","Text":"If x is minus 1,"},{"Start":"04:04.820 ","End":"04:08.270","Text":"1/2^minus 1 is 1 over 1/2,"},{"Start":"04:08.270 ","End":"04:11.540","Text":"which is 2, which gives us the mirror image,"},{"Start":"04:11.540 ","End":"04:13.835","Text":"in fact, of this point."},{"Start":"04:13.835 ","End":"04:16.595","Text":"Likewise, if x equals minus 2,"},{"Start":"04:16.595 ","End":"04:19.715","Text":"1/2^minus 2 is 4."},{"Start":"04:19.715 ","End":"04:21.875","Text":"If I try to sketch this,"},{"Start":"04:21.875 ","End":"04:23.705","Text":"we see it\u0027s decreasing."},{"Start":"04:23.705 ","End":"04:26.090","Text":"I can say a bit more about the parameter a,"},{"Start":"04:26.090 ","End":"04:30.545","Text":"other than the fact that it determines whether we\u0027re increasing or decreasing."},{"Start":"04:30.545 ","End":"04:34.270","Text":"As a increases, the graph just become steeper."},{"Start":"04:34.270 ","End":"04:37.715","Text":"For example, if I take a equals 4,"},{"Start":"04:37.715 ","End":"04:42.010","Text":"and I look at y equals 4^x."},{"Start":"04:42.010 ","End":"04:43.740","Text":"Then when x is 0,"},{"Start":"04:43.740 ","End":"04:45.060","Text":"y is still 1."},{"Start":"04:45.060 ","End":"04:46.450","Text":"But when x is 1,"},{"Start":"04:46.450 ","End":"04:49.610","Text":"y is already equal to 4."},{"Start":"04:49.610 ","End":"04:53.040","Text":"If x is minus 1, y is 1/4."},{"Start":"04:53.040 ","End":"04:55.745","Text":"We will get a much steeper graph,"},{"Start":"04:55.745 ","End":"04:58.505","Text":"goes down to 0 quicker over here."},{"Start":"04:58.505 ","End":"05:00.425","Text":"Similarly, on the other side,"},{"Start":"05:00.425 ","End":"05:05.930","Text":"if I let y equals 1/4^x,"},{"Start":"05:05.930 ","End":"05:07.925","Text":"as a gets smaller,"},{"Start":"05:07.925 ","End":"05:11.030","Text":"it gets steeper in the other way,"},{"Start":"05:11.030 ","End":"05:12.560","Text":"when x is minus 1,"},{"Start":"05:12.560 ","End":"05:14.300","Text":"then y is 4."},{"Start":"05:14.300 ","End":"05:16.235","Text":"It always goes through here."},{"Start":"05:16.235 ","End":"05:18.065","Text":"At 1, it becomes 1/4."},{"Start":"05:18.065 ","End":"05:20.170","Text":"We get something like this."},{"Start":"05:20.170 ","End":"05:23.390","Text":"In other words, a determines not only"},{"Start":"05:23.390 ","End":"05:27.080","Text":"which way whether it\u0027s increasing or decreasing, but how steeply."},{"Start":"05:27.080 ","End":"05:32.845","Text":"I would like to make some observations about the exponential function in general."},{"Start":"05:32.845 ","End":"05:35.600","Text":"1 observation is, if you look at this,"},{"Start":"05:35.600 ","End":"05:38.030","Text":"at the graphs, they\u0027re all positive."},{"Start":"05:38.030 ","End":"05:40.100","Text":"They\u0027re always above the x-axis."},{"Start":"05:40.100 ","End":"05:42.800","Text":"I didn\u0027t even need this bit of the drawing."},{"Start":"05:42.800 ","End":"05:44.525","Text":"It\u0027s always positive."},{"Start":"05:44.525 ","End":"05:46.510","Text":"Because it\u0027s always positive,"},{"Start":"05:46.510 ","End":"05:49.670","Text":"it\u0027s not going to intersect the x axis."},{"Start":"05:49.670 ","End":"05:52.370","Text":"There is no x-intercept."},{"Start":"05:52.370 ","End":"05:54.320","Text":"As for the y-intercept,"},{"Start":"05:54.320 ","End":"05:57.200","Text":"the y-intercept is always the same."},{"Start":"05:57.200 ","End":"06:01.370","Text":"It\u0027s always equal to 1 when x is 0, a^0 is 1."},{"Start":"06:01.370 ","End":"06:04.265","Text":"Or we can just say it\u0027s always 0,1,"},{"Start":"06:04.265 ","End":"06:06.235","Text":"which is this point here."},{"Start":"06:06.235 ","End":"06:09.350","Text":"There is 1 last thing I\u0027d like to mention that before we go"},{"Start":"06:09.350 ","End":"06:12.739","Text":"on to the next section on logarithmic functions,"},{"Start":"06:12.739 ","End":"06:16.130","Text":"is that there is a very special exponential function."},{"Start":"06:16.130 ","End":"06:21.425","Text":"It is the function y equals e^x."},{"Start":"06:21.425 ","End":"06:25.480","Text":"Some of you may have heard of the number e and some of you may not."},{"Start":"06:25.480 ","End":"06:27.440","Text":"But e, although it\u0027s a letter,"},{"Start":"06:27.440 ","End":"06:29.135","Text":"is a constant number,"},{"Start":"06:29.135 ","End":"06:35.640","Text":"pretty much like Pi is a number which is approximately equal to 3.1416."},{"Start":"06:36.370 ","End":"06:39.650","Text":"E is another special number in mathematics,"},{"Start":"06:39.650 ","End":"06:43.860","Text":"and e is approximately 2.718."},{"Start":"06:45.230 ","End":"06:51.005","Text":"It goes on, and you don\u0027t have to know exactly what it is or where it comes from."},{"Start":"06:51.005 ","End":"06:53.225","Text":"The calculator knows about it."},{"Start":"06:53.225 ","End":"06:55.370","Text":"You just have to know it\u0027s a special number."},{"Start":"06:55.370 ","End":"06:59.510","Text":"Why we use this number e. When you get down to calculus,"},{"Start":"06:59.510 ","End":"07:03.920","Text":"many formulae for integration and differentiation,"},{"Start":"07:03.920 ","End":"07:06.800","Text":"that everything comes up much simpler later."},{"Start":"07:06.800 ","End":"07:08.275","Text":"Perhaps after logarithms,"},{"Start":"07:08.275 ","End":"07:11.720","Text":"I\u0027ll explain what this number e is and where it comes from."},{"Start":"07:11.720 ","End":"07:14.300","Text":"That\u0027s all I\u0027ll say on that matter for now."},{"Start":"07:14.300 ","End":"07:17.280","Text":"Next up, logarithmic functions."}],"ID":9013},{"Watched":false,"Name":"The Logarithmic Function","Duration":"7m 41s","ChapterTopicVideoID":8822,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.989","Text":"In this clip, we\u0027ll talk a bit about the logarithmic function."},{"Start":"00:03.989 ","End":"00:11.520","Text":"This is a function of the form f of x equals log to the base of some number a,"},{"Start":"00:11.520 ","End":"00:16.080","Text":"sum parameter constant logarithm of x."},{"Start":"00:16.080 ","End":"00:18.960","Text":"But there are some restrictions on a,"},{"Start":"00:18.960 ","End":"00:22.555","Text":"just like there were restrictions on a in the exponential function."},{"Start":"00:22.555 ","End":"00:32.055","Text":"Here as there we have to have a positive and also we disqualify the value 1."},{"Start":"00:32.055 ","End":"00:36.555","Text":"There was another restriction on the domain of definition."},{"Start":"00:36.555 ","End":"00:40.945","Text":"We require that x be positive."},{"Start":"00:40.945 ","End":"00:44.015","Text":"You can only take the logarithm of a positive number."},{"Start":"00:44.015 ","End":"00:51.770","Text":"Some examples would be y equals log to the base 2 of x."},{"Start":"00:51.770 ","End":"00:59.735","Text":"You could have y equals log to the base 1/3 of x."},{"Start":"00:59.735 ","End":"01:04.430","Text":"You could have f of x equals log base,"},{"Start":"01:04.430 ","End":"01:07.700","Text":"I don\u0027t know, 4.7 of x and so on."},{"Start":"01:07.700 ","End":"01:10.550","Text":"But we must remember not to use the forbidden values,"},{"Start":"01:10.550 ","End":"01:14.975","Text":"not to put anything negative and not to use the value 1."},{"Start":"01:14.975 ","End":"01:18.020","Text":"Also, notice that there\u0027s only 1 parameter."},{"Start":"01:18.020 ","End":"01:22.790","Text":"We have just one parameter a determines which logarithmic function we have."},{"Start":"01:22.790 ","End":"01:24.950","Text":"This is in contrast to say,"},{"Start":"01:24.950 ","End":"01:27.739","Text":"the quadratic function where you have 3 parameters,"},{"Start":"01:27.739 ","End":"01:33.170","Text":"a, b, and c. We\u0027d like to see what the significance of a is."},{"Start":"01:33.170 ","End":"01:36.964","Text":"The main significance is in the shape of the graph."},{"Start":"01:36.964 ","End":"01:39.875","Text":"If a is bigger than 1,"},{"Start":"01:39.875 ","End":"01:42.900","Text":"then f is increasing,"},{"Start":"01:42.900 ","End":"01:45.545","Text":"f meaning this function is increasing."},{"Start":"01:45.545 ","End":"01:48.800","Text":"If a is between 0 and 1,"},{"Start":"01:48.800 ","End":"01:51.965","Text":"I mean the only possibility is here are between"},{"Start":"01:51.965 ","End":"01:55.540","Text":"either greater than 1 or between 0 and 1,"},{"Start":"01:55.540 ","End":"01:59.315","Text":"then if f is the opposite, it\u0027s decreasing."},{"Start":"01:59.315 ","End":"02:02.210","Text":"Let\u0027s draw the sketch of the graph."},{"Start":"02:02.210 ","End":"02:03.740","Text":"We could make a table."},{"Start":"02:03.740 ","End":"02:05.559","Text":"Let\u0027s take an example."},{"Start":"02:05.559 ","End":"02:10.050","Text":"For a greater than 1 will take a equals 2."},{"Start":"02:10.050 ","End":"02:15.950","Text":"Let\u0027s make a table for the case where a is equal to 2."},{"Start":"02:15.950 ","End":"02:23.180","Text":"In other words, the function is f of x equals log to the base 2 of x."},{"Start":"02:23.180 ","End":"02:25.250","Text":"We\u0027re only going to take x as positive."},{"Start":"02:25.250 ","End":"02:28.220","Text":"That\u0027s also why I drew this thing so short here."},{"Start":"02:28.220 ","End":"02:31.055","Text":"The domain remember was x bigger than 0."},{"Start":"02:31.055 ","End":"02:33.020","Text":"There\u0027s not going to be anything here,"},{"Start":"02:33.020 ","End":"02:34.895","Text":"It\u0027s going to be just empty."},{"Start":"02:34.895 ","End":"02:38.060","Text":"If we take a equals 2,"},{"Start":"02:38.060 ","End":"02:41.445","Text":"when x equals, let\u0027s say 1,"},{"Start":"02:41.445 ","End":"02:44.295","Text":"the log of 1 is 0."},{"Start":"02:44.295 ","End":"02:45.975","Text":"That\u0027s true for any a."},{"Start":"02:45.975 ","End":"02:51.110","Text":"In fact, all the graphs of log of x wherever a is going to pass through here,"},{"Start":"02:51.110 ","End":"02:52.390","Text":"there\u0027s going to be a special point,"},{"Start":"02:52.390 ","End":"02:54.065","Text":"meeting point of all the graphs."},{"Start":"02:54.065 ","End":"02:56.470","Text":"If x is 2,"},{"Start":"02:56.470 ","End":"03:00.155","Text":"log to the base 2 of 2 is 1."},{"Start":"03:00.155 ","End":"03:03.525","Text":"If it\u0027s 4 log to the base 2 of 4 is 2,"},{"Start":"03:03.525 ","End":"03:07.500","Text":"log to the base 2 of 8 is 3."},{"Start":"03:07.500 ","End":"03:12.000","Text":"We can go a bit in the other direction of log of 1/2,"},{"Start":"03:12.000 ","End":"03:14.310","Text":"because it\u0027s 2^minus 1,"},{"Start":"03:14.310 ","End":"03:16.605","Text":"the log would be minus 1."},{"Start":"03:16.605 ","End":"03:25.485","Text":"If we took x as a 1/4 log to the base 2 of a 1/4 is minus 2 because 2^minus 2 is a 1/4."},{"Start":"03:25.485 ","End":"03:28.514","Text":"Let me do a freehand sketch."},{"Start":"03:28.514 ","End":"03:34.130","Text":"Through here, here, here, here, here,"},{"Start":"03:34.130 ","End":"03:35.550","Text":"and so on,"},{"Start":"03:35.550 ","End":"03:38.270","Text":"up to here very crude rough sketch,"},{"Start":"03:38.270 ","End":"03:41.119","Text":"you get the idea though it\u0027s an increasing function."},{"Start":"03:41.119 ","End":"03:44.810","Text":"We go up and up,"},{"Start":"03:44.810 ","End":"03:48.365","Text":"and up and up, and so on."},{"Start":"03:48.365 ","End":"03:50.870","Text":"Now let\u0027s take an example of the other case."},{"Start":"03:50.870 ","End":"03:55.400","Text":"If I take a equals 1/2 and make a table,"},{"Start":"03:55.400 ","End":"03:59.300","Text":"we have 1/2 is 1."},{"Start":"03:59.300 ","End":"04:03.170","Text":"In fact, it\u0027s going to be the mirror image of the previous one,"},{"Start":"04:03.170 ","End":"04:05.135","Text":"the mirror image about the x-axis."},{"Start":"04:05.135 ","End":"04:07.670","Text":"Try a freehand sketch."},{"Start":"04:07.670 ","End":"04:11.150","Text":"A basically determines which way the graph"},{"Start":"04:11.150 ","End":"04:14.600","Text":"goes increasing or decreasing bigger than 1, less than 1."},{"Start":"04:14.600 ","End":"04:18.980","Text":"I can say a little bit more that if a gets even bigger,"},{"Start":"04:18.980 ","End":"04:21.214","Text":"say I took a equals 3,"},{"Start":"04:21.214 ","End":"04:24.425","Text":"then it would be shallower, pass through here."},{"Start":"04:24.425 ","End":"04:26.660","Text":"But it would be shallower."},{"Start":"04:26.660 ","End":"04:30.405","Text":"For example, the point 3,1 would be on it."},{"Start":"04:30.405 ","End":"04:32.855","Text":"If we took as the last example,"},{"Start":"04:32.855 ","End":"04:35.060","Text":"a equals 1/3,"},{"Start":"04:35.060 ","End":"04:38.300","Text":"we\u0027d actually get the mirror image of this one and it would"},{"Start":"04:38.300 ","End":"04:41.775","Text":"look something like they start somewhere here."},{"Start":"04:41.775 ","End":"04:45.309","Text":"It would still go through this central point"},{"Start":"04:45.309 ","End":"04:49.510","Text":"and just be a bit less deeper, something like this."},{"Start":"04:49.510 ","End":"04:53.810","Text":"That would be for a equals 1/3."},{"Start":"04:53.920 ","End":"04:56.860","Text":"Not only determines increasing, decreasing,"},{"Start":"04:56.860 ","End":"05:00.640","Text":"but how steeply, and you can see by the numbers which direction it goes in."},{"Start":"05:00.640 ","End":"05:03.970","Text":"What I would like to add is that mostly we won\u0027t"},{"Start":"05:03.970 ","End":"05:07.480","Text":"be concerning ourselves with all these different values of a."},{"Start":"05:07.480 ","End":"05:15.355","Text":"There\u0027s one special value of a that is used most commonly in mathematics for calculus."},{"Start":"05:15.355 ","End":"05:18.445","Text":"That number, it\u0027s when a equals e,"},{"Start":"05:18.445 ","End":"05:27.815","Text":"by which I mean that the function would be f of x equals log to the base e of x."},{"Start":"05:27.815 ","End":"05:30.770","Text":"We already mentioned e that it\u0027s just a constant,"},{"Start":"05:30.770 ","End":"05:34.700","Text":"a number that happens to equal 2.718,"},{"Start":"05:34.700 ","End":"05:39.845","Text":"approximately, because it falls into the category of bigger than 1."},{"Start":"05:39.845 ","End":"05:41.435","Text":"It\u0027s increasing."},{"Start":"05:41.435 ","End":"05:42.770","Text":"it\u0027s one of these."},{"Start":"05:42.770 ","End":"05:45.830","Text":"In fact, because e is between 2 and 3,"},{"Start":"05:45.830 ","End":"05:50.300","Text":"the e^x function would be something like this."},{"Start":"05:50.300 ","End":"05:53.885","Text":"That would be where a equals e,"},{"Start":"05:53.885 ","End":"06:03.080","Text":"which means that this is the function y equals log to the base e of x, the red one."},{"Start":"06:03.080 ","End":"06:06.289","Text":"Because it\u0027s so useful and so special,"},{"Start":"06:06.289 ","End":"06:08.465","Text":"it\u0027s given a special name,"},{"Start":"06:08.465 ","End":"06:16.780","Text":"the logarithm to the base e is given the name natural log, which is ln."},{"Start":"06:16.780 ","End":"06:19.730","Text":"We write natural log of x."},{"Start":"06:19.730 ","End":"06:21.410","Text":"When you see ln,"},{"Start":"06:21.410 ","End":"06:22.700","Text":"it just means the log."},{"Start":"06:22.700 ","End":"06:24.695","Text":"But to the special base e,"},{"Start":"06:24.695 ","End":"06:30.350","Text":"all the usual rules of logarithms apply to the natural log because e is just some number."},{"Start":"06:30.350 ","End":"06:38.195","Text":"I\u0027m talking about all these logarithm rules such as the natural log of 1 is equal to 0,"},{"Start":"06:38.195 ","End":"06:43.255","Text":"such that the natural log of x1 times x2,"},{"Start":"06:43.255 ","End":"06:49.085","Text":"and that will be the natural log of x1 plus natural log of x2."},{"Start":"06:49.085 ","End":"06:51.995","Text":"All the usual laws of the logarithms."},{"Start":"06:51.995 ","End":"07:01.910","Text":"Natural log of x1 over x2 is natural log of x1 minus the natural log of x2."},{"Start":"07:01.910 ","End":"07:05.195","Text":"Also continuing over here with specific numbers."},{"Start":"07:05.195 ","End":"07:07.325","Text":"Natural log of 1 is 0."},{"Start":"07:07.325 ","End":"07:15.540","Text":"The natural log say of e would equal 1 because e to the 1 is e. In general,"},{"Start":"07:15.540 ","End":"07:18.390","Text":"natural log of e^n,"},{"Start":"07:18.390 ","End":"07:20.630","Text":"n is usually a whole number,"},{"Start":"07:20.630 ","End":"07:26.750","Text":"but it also could be a fraction if that this is equal to n natural log of e, which is 1."},{"Start":"07:26.750 ","End":"07:32.755","Text":"This is just equal to n. In short all the usual rules of logarithms apply,"},{"Start":"07:32.755 ","End":"07:35.750","Text":"and I don\u0027t think there\u0027s anything more I want to say about"},{"Start":"07:35.750 ","End":"07:41.040","Text":"the natural logarithm or about the logarithms. So we\u0027re done."}],"ID":9014},{"Watched":false,"Name":"Basic Functions Part 1","Duration":"6m 57s","ChapterTopicVideoID":9300,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this clip, I\u0027d like to talk about some basic functions."},{"Start":"00:03.945 ","End":"00:09.300","Text":"The idea is to build up a stock of a few basic functions that you"},{"Start":"00:09.300 ","End":"00:14.894","Text":"should know how they look even without having to sketch from scratch to plot points."},{"Start":"00:14.894 ","End":"00:18.270","Text":"What we\u0027re going to start with in our collection will be,"},{"Start":"00:18.270 ","End":"00:19.620","Text":"point number 1,"},{"Start":"00:19.620 ","End":"00:23.550","Text":"functions like y equals x squared,"},{"Start":"00:23.550 ","End":"00:26.640","Text":"y equals x to the 4th,"},{"Start":"00:26.640 ","End":"00:29.385","Text":"then x to the 6th, and so on."},{"Start":"00:29.385 ","End":"00:34.710","Text":"In general, we\u0027re going to talk about y equals x to the n,"},{"Start":"00:34.710 ","End":"00:38.295","Text":"where n is even,"},{"Start":"00:38.295 ","End":"00:40.125","Text":"bigger or equal to 2."},{"Start":"00:40.125 ","End":"00:41.895","Text":"First 1 in the series,"},{"Start":"00:41.895 ","End":"00:44.614","Text":"x squared is a basic parabola."},{"Start":"00:44.614 ","End":"00:46.640","Text":"You should be familiar with that."},{"Start":"00:46.640 ","End":"00:51.740","Text":"I\u0027ll tell you why, x squared you could very quickly just make a set of axis,"},{"Start":"00:51.740 ","End":"00:54.770","Text":"or u-shape, should be symmetrical,"},{"Start":"00:54.770 ","End":"00:57.865","Text":"should have gone through 0,0 but you know the general shape."},{"Start":"00:57.865 ","End":"01:00.150","Text":"What you don\u0027t know maybe is x to the 4th,"},{"Start":"01:00.150 ","End":"01:01.675","Text":"x to the 6th, and so on."},{"Start":"01:01.675 ","End":"01:04.570","Text":"In general, they all look very similar,"},{"Start":"01:04.570 ","End":"01:06.785","Text":"with a little bit of difference."},{"Start":"01:06.785 ","End":"01:11.300","Text":"The main difference, if you take x to the 4th, for example,"},{"Start":"01:11.300 ","End":"01:16.595","Text":"y equals x to the 4th will look remarkably similar due to the general u-shape."},{"Start":"01:16.595 ","End":"01:18.020","Text":"It looks very much like this,"},{"Start":"01:18.020 ","End":"01:20.870","Text":"the main difference being that it goes up steeper,"},{"Start":"01:20.870 ","End":"01:24.170","Text":"so it will be closer to the y-axis and at the bottom,"},{"Start":"01:24.170 ","End":"01:26.240","Text":"it\u0027s a bit flatter, and so on."},{"Start":"01:26.240 ","End":"01:28.910","Text":"If I take x to the 6th or any of the higher ones,"},{"Start":"01:28.910 ","End":"01:31.415","Text":"they\u0027ll all have the same basic shape."},{"Start":"01:31.415 ","End":"01:33.930","Text":"They\u0027ll all go up steeply,"},{"Start":"01:33.930 ","End":"01:38.685","Text":"here a bit flatter but still going from 0,0 and ups steeply again."},{"Start":"01:38.685 ","End":"01:41.265","Text":"That\u0027s in general y equals x to the n,"},{"Start":"01:41.265 ","End":"01:47.205","Text":"where n is even and greater or equal to 2 and it should go through the 0,0."},{"Start":"01:47.205 ","End":"01:49.165","Text":"Now let\u0027s move on to the next set."},{"Start":"01:49.165 ","End":"01:52.745","Text":"We\u0027re going to talk about the odd ones instead of the even ones."},{"Start":"01:52.745 ","End":"01:57.275","Text":"This time, it\u0027s going to be y equals x cubed,"},{"Start":"01:57.275 ","End":"02:00.290","Text":"y equals x to the 5th,"},{"Start":"02:00.290 ","End":"02:03.355","Text":"y equals x to the 7th, and so on."},{"Start":"02:03.355 ","End":"02:05.655","Text":"y equals x to the n,"},{"Start":"02:05.655 ","End":"02:09.240","Text":"where n is odd, 3, 5,"},{"Start":"02:09.240 ","End":"02:10.680","Text":"7, 9, etc,"},{"Start":"02:10.680 ","End":"02:13.505","Text":"and bigger or equal to 3."},{"Start":"02:13.505 ","End":"02:14.750","Text":"The very first 1,"},{"Start":"02:14.750 ","End":"02:16.175","Text":"y equals x cubed,"},{"Start":"02:16.175 ","End":"02:18.890","Text":"you\u0027re probably familiar with."},{"Start":"02:18.890 ","End":"02:23.600","Text":"x cubed goes like this, flattens out here,"},{"Start":"02:23.600 ","End":"02:26.815","Text":"and then it rises again steeply,"},{"Start":"02:26.815 ","End":"02:28.830","Text":"y equals x cubed."},{"Start":"02:28.830 ","End":"02:33.169","Text":"Note also that when x is 0, y is 0."},{"Start":"02:33.169 ","End":"02:36.065","Text":"You can tell this equation."},{"Start":"02:36.065 ","End":"02:38.995","Text":"Also that when x is 1,"},{"Start":"02:38.995 ","End":"02:40.590","Text":"1 to the n is 1,"},{"Start":"02:40.590 ","End":"02:44.040","Text":"and likewise minus 1 to the n is minus 1,"},{"Start":"02:44.040 ","End":"02:45.285","Text":"provided n is odd."},{"Start":"02:45.285 ","End":"02:48.795","Text":"These 3 points appear on all this family."},{"Start":"02:48.795 ","End":"02:52.185","Text":"I\u0027m going to show you the x to the 5th next,"},{"Start":"02:52.185 ","End":"02:55.125","Text":"and in general x to the n. As you can see,"},{"Start":"02:55.125 ","End":"02:58.560","Text":"they all look very similar, the x to the 5th."},{"Start":"02:58.560 ","End":"03:02.580","Text":"It will also go through these 3 points that I wrote here."},{"Start":"03:02.580 ","End":"03:05.940","Text":"It just goes up steeper because it\u0027s x to the 5th."},{"Start":"03:05.940 ","End":"03:08.835","Text":"For example, if x is 2,"},{"Start":"03:08.835 ","End":"03:10.410","Text":"here it\u0027s 8,"},{"Start":"03:10.410 ","End":"03:13.125","Text":"2 cubed, but here it\u0027s 32."},{"Start":"03:13.125 ","End":"03:16.260","Text":"In general, x to the n as n grows larger,"},{"Start":"03:16.260 ","End":"03:20.720","Text":"it will be higher up and therefore it will be narrower and closer to the axis."},{"Start":"03:20.720 ","End":"03:23.345","Text":"But the general picture is actually the same."},{"Start":"03:23.345 ","End":"03:25.565","Text":"You just draw a curved line,"},{"Start":"03:25.565 ","End":"03:28.270","Text":"goes through the origin and upwards."},{"Start":"03:28.270 ","End":"03:32.040","Text":"If you can manage it, it\u0027s more correct to make it flat here,"},{"Start":"03:32.040 ","End":"03:34.640","Text":"it flattens out, and then it gets steeper,"},{"Start":"03:34.640 ","End":"03:36.800","Text":"flattens out, gets steeper."},{"Start":"03:36.800 ","End":"03:38.660","Text":"Onto the next 1."},{"Start":"03:38.660 ","End":"03:43.400","Text":"In part 3, we\u0027ll continue with y equals x to the"},{"Start":"03:43.400 ","End":"03:48.405","Text":"negative 1. y equals x to the negative 3,"},{"Start":"03:48.405 ","End":"03:53.150","Text":"y equals x to the negative 5, etc."},{"Start":"03:53.150 ","End":"03:57.715","Text":"In general, y equals x to the n,"},{"Start":"03:57.715 ","End":"04:00.000","Text":"but n gets negative values."},{"Start":"04:00.000 ","End":"04:01.700","Text":"n is equal to minus 1,"},{"Start":"04:01.700 ","End":"04:04.640","Text":"minus 3, minus 5, etc."},{"Start":"04:04.640 ","End":"04:08.630","Text":"I could have just said n is a negative odd number."},{"Start":"04:08.630 ","End":"04:14.610","Text":"Let\u0027s take the first 1. x to the minus 1 is just 1 over x."},{"Start":"04:14.610 ","End":"04:18.090","Text":"What we need is y equals 1 over x."},{"Start":"04:18.090 ","End":"04:19.980","Text":"You may not be familiar with this."},{"Start":"04:19.980 ","End":"04:24.350","Text":"This time let\u0027s make a little table and we\u0027ll take convenient values."},{"Start":"04:24.350 ","End":"04:28.900","Text":"For example, it\u0027s easy to take x equals 1 and y equals 1."},{"Start":"04:28.900 ","End":"04:34.800","Text":"We want something a bit less than 1 also so let\u0027s try 0.5. y is 1 over 0.5 is 2,"},{"Start":"04:34.800 ","End":"04:39.390","Text":"and the reverse also effect is 2, y is 0.5."},{"Start":"04:39.390 ","End":"04:41.235","Text":"Also want to try some negatives."},{"Start":"04:41.235 ","End":"04:46.545","Text":"If we have minus 0.5 here or over minus 0.5 will be minus 2,"},{"Start":"04:46.545 ","End":"04:47.850","Text":"x is minus 1,"},{"Start":"04:47.850 ","End":"04:49.290","Text":"y is minus 1."},{"Start":"04:49.290 ","End":"04:50.910","Text":"If x is minus 2,"},{"Start":"04:50.910 ","End":"04:53.580","Text":"y is minus 0.5."},{"Start":"04:53.580 ","End":"04:55.920","Text":"If I plot all these,"},{"Start":"04:55.920 ","End":"05:00.014","Text":"then I\u0027ll get something like this where these are the points I computed,"},{"Start":"05:00.014 ","End":"05:05.235","Text":"1,1 and 2, 0.5, and so on."},{"Start":"05:05.235 ","End":"05:07.275","Text":"y minus 1 minus 1."},{"Start":"05:07.275 ","End":"05:10.290","Text":"Actually, it turns out that these 2,"},{"Start":"05:10.290 ","End":"05:15.135","Text":"if I put in the 1,1 and this 1 here,"},{"Start":"05:15.135 ","End":"05:16.795","Text":"minus 1, minus 1,"},{"Start":"05:16.795 ","End":"05:19.895","Text":"these actually appear in the whole series."},{"Start":"05:19.895 ","End":"05:21.725","Text":"If I drew the next 1,"},{"Start":"05:21.725 ","End":"05:25.835","Text":"y equals x to the power of minus 3,"},{"Start":"05:25.835 ","End":"05:31.880","Text":"we get very similar except that it adheres more closely to the axis."},{"Start":"05:31.880 ","End":"05:34.580","Text":"It clings closer to the axis over here."},{"Start":"05:34.580 ","End":"05:37.130","Text":"But of course, it still goes through these 2 points."},{"Start":"05:37.130 ","End":"05:38.750","Text":"All of them go through 1,"},{"Start":"05:38.750 ","End":"05:42.230","Text":"1, and minus 1, minus 1."},{"Start":"05:42.230 ","End":"05:45.440","Text":"Then they just go closer to the axes more quickly."},{"Start":"05:45.440 ","End":"05:48.140","Text":"This could be 1 over x to the 3,"},{"Start":"05:48.140 ","End":"05:49.490","Text":"or it could be in general,"},{"Start":"05:49.490 ","End":"05:54.025","Text":"1 over x to the nth or some n in the series we talked about."},{"Start":"05:54.025 ","End":"05:58.055","Text":"Something I almost forgot to mention about this family of functions,"},{"Start":"05:58.055 ","End":"06:00.815","Text":"they don\u0027t intersect the axis."},{"Start":"06:00.815 ","End":"06:04.175","Text":"Let\u0027s use this 1 to represent the whole x to the n,"},{"Start":"06:04.175 ","End":"06:06.725","Text":"not just 1 over x or 1 over x cubed."},{"Start":"06:06.725 ","End":"06:12.050","Text":"Notice that it can\u0027t intersect the y-axis because to the y-axis,"},{"Start":"06:12.050 ","End":"06:15.260","Text":"if I\u0027m looking for the y-intercepts to the y-axis,"},{"Start":"06:15.260 ","End":"06:18.575","Text":"then I have to substitute x equals 0."},{"Start":"06:18.575 ","End":"06:22.630","Text":"I get y equals 1 over 0 to the n,"},{"Start":"06:22.630 ","End":"06:25.095","Text":"and that\u0027s not possible, it\u0027s 1 over 0."},{"Start":"06:25.095 ","End":"06:27.165","Text":"It doesn\u0027t intersect the y-axis."},{"Start":"06:27.165 ","End":"06:28.920","Text":"As to the x-axis,"},{"Start":"06:28.920 ","End":"06:30.240","Text":"or the x-intercept,"},{"Start":"06:30.240 ","End":"06:32.220","Text":"then I have to put y equals 0."},{"Start":"06:32.220 ","End":"06:34.665","Text":"If I put y equals 0,"},{"Start":"06:34.665 ","End":"06:44.955","Text":"then I get 0 equals 1 over x to the n. If I multiply both sides by x to the n, I get 0."},{"Start":"06:44.955 ","End":"06:47.475","Text":"This gives me 0 equals 1,"},{"Start":"06:47.475 ","End":"06:50.160","Text":"which is certainly not possible."},{"Start":"06:50.160 ","End":"06:54.735","Text":"No x-intercepts, no y-intercepts to this family of functions."},{"Start":"06:54.735 ","End":"06:57.940","Text":"Now can move on to the next set."}],"ID":9612},{"Watched":false,"Name":"Basic Functions Part 2","Duration":"7m 50s","ChapterTopicVideoID":9301,"CourseChapterTopicPlaylistID":1179,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"Now we come to the next set of basic functions."},{"Start":"00:03.900 ","End":"00:06.585","Text":"These are the functions as written here,"},{"Start":"00:06.585 ","End":"00:10.005","Text":"x^minus 2, x^minus 4, x^minus 6."},{"Start":"00:10.005 ","End":"00:15.405","Text":"These are the x to the power of something that\u0027s negative and even."},{"Start":"00:15.405 ","End":"00:18.435","Text":"If it\u0027s convenient, we can use the other form,"},{"Start":"00:18.435 ","End":"00:21.900","Text":"like x^minus 2 is 1 over x squared,"},{"Start":"00:21.900 ","End":"00:24.165","Text":"and here we have 1 over x^4,"},{"Start":"00:24.165 ","End":"00:27.030","Text":"1 over x^6, and so on."},{"Start":"00:27.030 ","End":"00:30.225","Text":"They all look very similar in their basic shape."},{"Start":"00:30.225 ","End":"00:34.860","Text":"I\u0027ll show you they look like the 1 over x squared."},{"Start":"00:34.860 ","End":"00:37.680","Text":"Notice this could be 1 over x squared."},{"Start":"00:37.680 ","End":"00:39.820","Text":"They could be any one of them really."},{"Start":"00:39.820 ","End":"00:42.880","Text":"Notice it\u0027s always above the axis,"},{"Start":"00:42.880 ","End":"00:48.445","Text":"and that\u0027s because 1 over x squared is 1 over a positive number."},{"Start":"00:48.445 ","End":"00:50.040","Text":"That\u0027s always positive,"},{"Start":"00:50.040 ","End":"00:55.925","Text":"so the thing is always floating above the x-axis, just the domain."},{"Start":"00:55.925 ","End":"00:58.660","Text":"The domain x cannot be 0,"},{"Start":"00:58.660 ","End":"01:01.310","Text":"and so this is not included."},{"Start":"01:01.310 ","End":"01:03.350","Text":"It doesn\u0027t touch the y-axis."},{"Start":"01:03.350 ","End":"01:05.740","Text":"There is no y-intercept."},{"Start":"01:05.740 ","End":"01:08.060","Text":"Like I said, there\u0027s no x-intercept"},{"Start":"01:08.060 ","End":"01:10.940","Text":"either because this thing is always strictly positive,"},{"Start":"01:10.940 ","End":"01:13.220","Text":"1 over something can\u0027t even be 0."},{"Start":"01:13.220 ","End":"01:17.480","Text":"I do want to say a word about how they do differ,"},{"Start":"01:17.480 ","End":"01:20.600","Text":"even though this is supposed to represent all of them."},{"Start":"01:20.600 ","End":"01:24.470","Text":"Let\u0027s just say, although this does represent the family,"},{"Start":"01:24.470 ","End":"01:30.585","Text":"that this one is specifically y equals 1 over x squared."},{"Start":"01:30.585 ","End":"01:36.985","Text":"Let\u0027s superimpose y equals 1 over x^4."},{"Start":"01:36.985 ","End":"01:39.294","Text":"But I\u0027ll do that one in orange."},{"Start":"01:39.294 ","End":"01:45.415","Text":"The way it differs is that above 1,1, it\u0027s higher."},{"Start":"01:45.415 ","End":"01:47.490","Text":"It still goes through here,"},{"Start":"01:47.490 ","End":"01:49.730","Text":"and then below that part,"},{"Start":"01:49.730 ","End":"01:52.220","Text":"it actually goes to 0 quicker,"},{"Start":"01:52.220 ","End":"01:54.170","Text":"meaning it\u0027s lower than this one."},{"Start":"01:54.170 ","End":"01:58.340","Text":"Well, the picture explains what I mean, and likewise here."},{"Start":"01:58.340 ","End":"02:02.580","Text":"That\u0027s it for these negative-even powers."},{"Start":"02:02.580 ","End":"02:04.920","Text":"Onto the next set,"},{"Start":"02:04.920 ","End":"02:10.370","Text":"we\u0027re continuing with the x^n functions."},{"Start":"02:10.370 ","End":"02:14.615","Text":"This time the single function where n equals 1/2,"},{"Start":"02:14.615 ","End":"02:17.750","Text":"which gives us y equals square root of x."},{"Start":"02:17.750 ","End":"02:20.150","Text":"You may and may not be familiar with this."},{"Start":"02:20.150 ","End":"02:22.609","Text":"Why don\u0027t we even sketch it together?"},{"Start":"02:22.609 ","End":"02:24.980","Text":"We\u0027ll do the table and everything."},{"Start":"02:24.980 ","End":"02:31.070","Text":"Now, the domain of this function is x bigger or equal to 0 because the square root,"},{"Start":"02:31.070 ","End":"02:32.780","Text":"we can\u0027t take of a negative number."},{"Start":"02:32.780 ","End":"02:34.690","Text":"We can take 0,"},{"Start":"02:34.690 ","End":"02:36.720","Text":"and we can take positive,"},{"Start":"02:36.720 ","End":"02:41.935","Text":"so we\u0027re only going to get stuff on the right side of the y-axis."},{"Start":"02:41.935 ","End":"02:45.720","Text":"Notice that if I put in a positive number,"},{"Start":"02:45.720 ","End":"02:48.410","Text":"the square root is also positive."},{"Start":"02:48.410 ","End":"02:52.925","Text":"We\u0027re really only going to see stuff in the first quadrant."},{"Start":"02:52.925 ","End":"02:56.390","Text":"0 is the most convenient to start off with."},{"Start":"02:56.390 ","End":"03:00.050","Text":"That\u0027s actually the very first value we\u0027re allowed to substitute,"},{"Start":"03:00.050 ","End":"03:02.270","Text":"so y also is 0."},{"Start":"03:02.270 ","End":"03:03.960","Text":"It\u0027s the square root."},{"Start":"03:03.960 ","End":"03:06.425","Text":"1 is also easy to compute."},{"Start":"03:06.425 ","End":"03:07.970","Text":"Square root of 1 is 1."},{"Start":"03:07.970 ","End":"03:12.470","Text":"I\u0027m going to go for numbers that will give us nice, whole square roots."},{"Start":"03:12.470 ","End":"03:15.620","Text":"So 4 would be next and its square root would be 2,"},{"Start":"03:15.620 ","End":"03:19.400","Text":"and maybe we can fit into 9, which would be 3."},{"Start":"03:19.400 ","End":"03:21.150","Text":"Let\u0027s see if we have 0."},{"Start":"03:21.150 ","End":"03:23.475","Text":"We\u0027re going to need 1."},{"Start":"03:23.475 ","End":"03:26.835","Text":"Then 4 we\u0027ll have,"},{"Start":"03:26.835 ","End":"03:28.965","Text":"9 we\u0027re going to take."},{"Start":"03:28.965 ","End":"03:31.805","Text":"We\u0027ll need here 0, we have that."},{"Start":"03:31.805 ","End":"03:34.110","Text":"We\u0027ll need 1,"},{"Start":"03:34.110 ","End":"03:36.585","Text":"2, and 3."},{"Start":"03:36.585 ","End":"03:40.790","Text":"We\u0027ll get 0, 0, 1, 1,"},{"Start":"03:40.790 ","End":"03:43.730","Text":"4 gives us 2,"},{"Start":"03:43.730 ","End":"03:46.460","Text":"and 9 gives us 3,"},{"Start":"03:46.460 ","End":"03:47.780","Text":"which is somewhere here."},{"Start":"03:47.780 ","End":"03:50.765","Text":"It\u0027ll actually starts out vertically,"},{"Start":"03:50.765 ","End":"03:54.140","Text":"and then gradually gets flatter and flatter."},{"Start":"03:54.140 ","End":"03:57.020","Text":"Of course, it goes on to infinity."},{"Start":"03:57.020 ","End":"03:59.650","Text":"That\u0027s really all I have to say."},{"Start":"03:59.650 ","End":"04:04.905","Text":"Number 6, again in the series of the x^n."},{"Start":"04:04.905 ","End":"04:06.775","Text":"This time n equals 1/3,"},{"Start":"04:06.775 ","End":"04:09.080","Text":"which gives us the cube root of x."},{"Start":"04:09.080 ","End":"04:10.400","Text":"A little bit less famous,"},{"Start":"04:10.400 ","End":"04:13.655","Text":"but still useful to add to our stock of basic functions."},{"Start":"04:13.655 ","End":"04:16.340","Text":"It\u0027s a bit different than the square root of x."},{"Start":"04:16.340 ","End":"04:19.595","Text":"For one thing, its domain is all of x."},{"Start":"04:19.595 ","End":"04:21.960","Text":"Let\u0027s just make a sketch,"},{"Start":"04:21.960 ","End":"04:25.120","Text":"draw a table, plot the points."},{"Start":"04:25.430 ","End":"04:28.985","Text":"When x is 0,"},{"Start":"04:28.985 ","End":"04:31.940","Text":"the cube root of 0 is 0."},{"Start":"04:31.940 ","End":"04:33.860","Text":"Let\u0027s keep going with a positive."},{"Start":"04:33.860 ","End":"04:35.930","Text":"Let\u0027s try taking x equals 1."},{"Start":"04:35.930 ","End":"04:38.365","Text":"Cube root of 1 is 1."},{"Start":"04:38.365 ","End":"04:40.140","Text":"2, 3, 4, and so on,"},{"Start":"04:40.140 ","End":"04:43.475","Text":"a messy comes out a cube root, need the calculator."},{"Start":"04:43.475 ","End":"04:48.430","Text":"Let\u0027s go straight to 8 because cube root of 8 is a nice whole number,"},{"Start":"04:48.430 ","End":"04:51.420","Text":"comes out 2, and so on."},{"Start":"04:51.420 ","End":"04:53.480","Text":"To get to 3, we\u0027d have to get up to 27."},{"Start":"04:53.480 ","End":"04:54.950","Text":"I don\u0027t think I have room for that."},{"Start":"04:54.950 ","End":"04:57.455","Text":"But we go into the negative numbers."},{"Start":"04:57.455 ","End":"04:58.850","Text":"Exactly the same,"},{"Start":"04:58.850 ","End":"05:01.039","Text":"cube root of a negative is a negative."},{"Start":"05:01.039 ","End":"05:04.430","Text":"Take minus 1 and the cube root of that is minus 1."},{"Start":"05:04.430 ","End":"05:07.370","Text":"Remember minus times minus times minus is also minus,"},{"Start":"05:07.370 ","End":"05:09.215","Text":"so there\u0027s no problem with negatives here."},{"Start":"05:09.215 ","End":"05:16.425","Text":"Likewise, we could take minus 8 here and get minus 2. Let\u0027s plot those."},{"Start":"05:16.425 ","End":"05:18.020","Text":"Just like we did with the square root,"},{"Start":"05:18.020 ","End":"05:19.775","Text":"I\u0027d like to do it in red."},{"Start":"05:19.775 ","End":"05:24.055","Text":"So 0,0 gives us this point here."},{"Start":"05:24.055 ","End":"05:26.775","Text":"Let\u0027s say that this is 1,"},{"Start":"05:26.775 ","End":"05:28.860","Text":"so we get here 1,"},{"Start":"05:28.860 ","End":"05:31.665","Text":"1, that\u0027ll be 1."},{"Start":"05:31.665 ","End":"05:33.625","Text":"To get to 2,"},{"Start":"05:33.625 ","End":"05:35.150","Text":"we need to go all the way up to 8."},{"Start":"05:35.150 ","End":"05:36.710","Text":"I don\u0027t know exactly where that is,"},{"Start":"05:36.710 ","End":"05:39.065","Text":"but let\u0027s say that 8 is somewhere here."},{"Start":"05:39.065 ","End":"05:43.300","Text":"Then we get this point here, 8,2."},{"Start":"05:43.300 ","End":"05:45.170","Text":"This is actually an odd function,"},{"Start":"05:45.170 ","End":"05:48.320","Text":"so we get exactly the same on the other side rotated."},{"Start":"05:48.320 ","End":"05:49.745","Text":"For minus 1,"},{"Start":"05:49.745 ","End":"05:52.435","Text":"we get this point here say,"},{"Start":"05:52.435 ","End":"05:54.950","Text":"and get to minus 2,"},{"Start":"05:54.950 ","End":"05:57.380","Text":"we have to go all the way up to minus 8."},{"Start":"05:57.380 ","End":"05:59.165","Text":"Let\u0027s say that\u0027s here."},{"Start":"05:59.165 ","End":"06:02.930","Text":"Now we just have to connect them. Here it is."},{"Start":"06:02.930 ","End":"06:06.545","Text":"The cube root of x goes along like this,"},{"Start":"06:06.545 ","End":"06:08.870","Text":"then it starts going upwards."},{"Start":"06:08.870 ","End":"06:11.750","Text":"Actually, tangent is vertical at this point,"},{"Start":"06:11.750 ","End":"06:15.400","Text":"and then it gets flatter and flatter again and so on to infinity."},{"Start":"06:15.400 ","End":"06:21.785","Text":"The last one I\u0027d like to show you will be the absolute value of x coming up."},{"Start":"06:21.785 ","End":"06:24.575","Text":"Here we are with basic function number 7,"},{"Start":"06:24.575 ","End":"06:26.765","Text":"y equals absolute value of x."},{"Start":"06:26.765 ","End":"06:31.615","Text":"Very famous function you should certainly know about."},{"Start":"06:31.615 ","End":"06:34.170","Text":"Let\u0027s just do it with a table."},{"Start":"06:34.170 ","End":"06:35.980","Text":"Let\u0027s start right away."},{"Start":"06:35.980 ","End":"06:38.510","Text":"Notice that x can be anything."},{"Start":"06:38.510 ","End":"06:40.430","Text":"There\u0027s no restriction on the domain."},{"Start":"06:40.430 ","End":"06:44.445","Text":"Domain is all x because whether x is positive or negative,"},{"Start":"06:44.445 ","End":"06:47.050","Text":"absolute value of x is defined."},{"Start":"06:47.050 ","End":"06:48.850","Text":"But y will always be positive,"},{"Start":"06:48.850 ","End":"06:51.790","Text":"which is why I drew this piece shorter because I know a couple"},{"Start":"06:51.790 ","End":"06:55.000","Text":"of things are going to positive or at least non-negative."},{"Start":"06:55.000 ","End":"06:58.765","Text":"When x is 0, the absolute value is defined as 0."},{"Start":"06:58.765 ","End":"07:02.025","Text":"Then if we take 1 or minus 1,"},{"Start":"07:02.025 ","End":"07:03.975","Text":"the absolute value is going to be 1."},{"Start":"07:03.975 ","End":"07:06.570","Text":"If we take 2 or minus 2,"},{"Start":"07:06.570 ","End":"07:08.685","Text":"the absolute value is going to be 2."},{"Start":"07:08.685 ","End":"07:11.855","Text":"If we have 3 or minus 3,"},{"Start":"07:11.855 ","End":"07:14.665","Text":"then the absolute value is going to be 3."},{"Start":"07:14.665 ","End":"07:17.770","Text":"If I plot all these points, 0,0 is here,"},{"Start":"07:17.770 ","End":"07:19.705","Text":"1,1 would be here,"},{"Start":"07:19.705 ","End":"07:23.200","Text":"2,2 is here, 3,3 is here."},{"Start":"07:23.200 ","End":"07:25.175","Text":"Likewise, on the minus 1,"},{"Start":"07:25.175 ","End":"07:28.770","Text":"this is also 1, 2, 3."},{"Start":"07:28.770 ","End":"07:31.005","Text":"Minus 1 is also 1,"},{"Start":"07:31.005 ","End":"07:36.090","Text":"minus 2, 2, minus 3, 3."},{"Start":"07:36.090 ","End":"07:38.770","Text":"It\u0027s supposed to come out a straight line."},{"Start":"07:38.770 ","End":"07:41.120","Text":"That\u0027s the rough sketch. This is what it will look like."},{"Start":"07:41.120 ","End":"07:44.270","Text":"Of course, it will go on indefinitely to infinity."},{"Start":"07:44.270 ","End":"07:46.820","Text":"Make the introduction to absolute value of x,"},{"Start":"07:46.820 ","End":"07:51.450","Text":"and that\u0027s the last one in the set of basic functions we\u0027ll be learning about."}],"ID":9613}],"Thumbnail":null,"ID":1179},{"Name":"The Domain of Definition of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Domain of Definition of a Function","Duration":"4m 56s","ChapterTopicVideoID":8228,"CourseChapterTopicPlaylistID":1173,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this clip, we\u0027ll talk about the domain of definition of a function."},{"Start":"00:04.680 ","End":"00:08.460","Text":"Domain of definition sometimes is just called the domain."},{"Start":"00:08.460 ","End":"00:09.705","Text":"What does it mean?"},{"Start":"00:09.705 ","End":"00:12.150","Text":"Well, basically, when we have a function,"},{"Start":"00:12.150 ","End":"00:16.485","Text":"the domain of definition simply means the values of x,"},{"Start":"00:16.485 ","End":"00:19.710","Text":"for which it makes sense to say f of x."},{"Start":"00:19.710 ","End":"00:21.599","Text":"Or put it in other words,"},{"Start":"00:21.599 ","End":"00:26.050","Text":"which values of x are we allowed to substitute in the function f?"},{"Start":"00:26.050 ","End":"00:27.720","Text":"Let\u0027s take an example."},{"Start":"00:27.720 ","End":"00:36.030","Text":"The first example, we\u0027ll take f of x is 24 over 2 minus x."},{"Start":"00:36.030 ","End":"00:40.865","Text":"Let\u0027s see what values of x it makes sense to substitute here."},{"Start":"00:40.865 ","End":"00:44.255","Text":"For example, could we put x equals 0."},{"Start":"00:44.255 ","End":"00:46.715","Text":"2 minus 0 is 2,"},{"Start":"00:46.715 ","End":"00:48.260","Text":"24 over 2 is 12."},{"Start":"00:48.260 ","End":"00:50.580","Text":"Could we put x equals 1?"},{"Start":"00:50.580 ","End":"00:53.780","Text":"24 over 2 minus 1, 24."},{"Start":"00:53.780 ","End":"00:55.895","Text":"We put x equals 3."},{"Start":"00:55.895 ","End":"00:57.860","Text":"2 minus 3 is negative 1,"},{"Start":"00:57.860 ","End":"01:00.574","Text":"24 over negative 1 is minus 24."},{"Start":"01:00.574 ","End":"01:02.225","Text":"What can we put?"},{"Start":"01:02.225 ","End":"01:03.710","Text":"Well, if you remember,"},{"Start":"01:03.710 ","End":"01:06.480","Text":"we\u0027re not allowed to divide by 0 vector."},{"Start":"01:06.480 ","End":"01:08.990","Text":"That\u0027s the only number we\u0027re not allowed to divide by."},{"Start":"01:08.990 ","End":"01:10.910","Text":"Well, if we put x equals 2,"},{"Start":"01:10.910 ","End":"01:15.280","Text":"we\u0027ll get 24 over 0 and we\u0027re not allowed to divide by 0."},{"Start":"01:15.280 ","End":"01:17.145","Text":"That\u0027s the only problem that can be."},{"Start":"01:17.145 ","End":"01:21.110","Text":"Every value of x is okay except for x equals 2."},{"Start":"01:21.110 ","End":"01:25.280","Text":"We say that the domain of definition is everything except 2."},{"Start":"01:25.280 ","End":"01:28.445","Text":"Write simply as x not equal to 2."},{"Start":"01:28.445 ","End":"01:30.740","Text":"Now let\u0027s take another example."},{"Start":"01:30.740 ","End":"01:32.210","Text":"As our next example,"},{"Start":"01:32.210 ","End":"01:39.100","Text":"we\u0027ll have f of x is equal to the square root of x minus 1."},{"Start":"01:39.100 ","End":"01:42.080","Text":"Again, we start looking at values of x,"},{"Start":"01:42.080 ","End":"01:43.865","Text":"which we could put in here."},{"Start":"01:43.865 ","End":"01:49.175","Text":"We try x equals 2 and you get 2 minus 1 is 1,"},{"Start":"01:49.175 ","End":"01:51.600","Text":"square root of 1, that\u0027s fine."},{"Start":"01:51.600 ","End":"01:54.365","Text":"We could take x equals 5."},{"Start":"01:54.365 ","End":"01:57.379","Text":"5 minus 1 is 4. No problem."},{"Start":"01:57.379 ","End":"01:59.465","Text":"Could even take x equals 3."},{"Start":"01:59.465 ","End":"02:01.765","Text":"3 minus 1 is 2."},{"Start":"02:01.765 ","End":"02:03.000","Text":"Square root of 2,"},{"Start":"02:03.000 ","End":"02:04.070","Text":"it\u0027s not a whole number,"},{"Start":"02:04.070 ","End":"02:06.155","Text":"but I can say square root of 2."},{"Start":"02:06.155 ","End":"02:10.025","Text":"X equals 100, square root of 99."},{"Start":"02:10.025 ","End":"02:12.155","Text":"Let\u0027s try x equals 0,"},{"Start":"02:12.155 ","End":"02:16.280","Text":"square root of minus 1 seem to remember"},{"Start":"02:16.280 ","End":"02:21.140","Text":"that square roots you can only take for positive numbers and 0,"},{"Start":"02:21.140 ","End":"02:22.940","Text":"but you can\u0027t take for negative numbers."},{"Start":"02:22.940 ","End":"02:27.320","Text":"If x is 1, square root of 1 minus 1 square root of 0 is 0,"},{"Start":"02:27.320 ","End":"02:29.630","Text":"is okay, x is 1 is okay."},{"Start":"02:29.630 ","End":"02:31.670","Text":"X is 0 was not okay."},{"Start":"02:31.670 ","End":"02:33.275","Text":"X is to was okay."},{"Start":"02:33.275 ","End":"02:35.240","Text":"In general, if you remember,"},{"Start":"02:35.240 ","End":"02:38.360","Text":"when can we take the square root of something which helps"},{"Start":"02:38.360 ","End":"02:41.750","Text":"symbolically just a writer as a little box,"},{"Start":"02:41.750 ","End":"02:45.080","Text":"to take the square root of x or when does it make sense?"},{"Start":"02:45.080 ","End":"02:47.495","Text":"It\u0027s when we have a positive number."},{"Start":"02:47.495 ","End":"02:49.100","Text":"But as I said, not just positive,"},{"Start":"02:49.100 ","End":"02:50.570","Text":"we can even have 0."},{"Start":"02:50.570 ","End":"02:52.370","Text":"Along with it\u0027s not negative,"},{"Start":"02:52.370 ","End":"02:56.135","Text":"user calls and says non negative numbers when it\u0027s bigger or equal to."},{"Start":"02:56.135 ","End":"03:00.255","Text":"In this case, x minus 1 has to be non negative."},{"Start":"03:00.255 ","End":"03:04.640","Text":"The domain of definition is x bigger or equal to"},{"Start":"03:04.640 ","End":"03:10.400","Text":"1 because that will give us x minus 1 is bigger or equal to 0 as here."},{"Start":"03:10.400 ","End":"03:12.005","Text":"There\u0027s another example."},{"Start":"03:12.005 ","End":"03:13.625","Text":"Let\u0027s take a third example."},{"Start":"03:13.625 ","End":"03:15.230","Text":"In this third example,"},{"Start":"03:15.230 ","End":"03:21.290","Text":"I\u0027m going to take the story from the previous clip and just in case you missed it,"},{"Start":"03:21.290 ","End":"03:24.125","Text":"that\u0027s where we drew a little house."},{"Start":"03:24.125 ","End":"03:27.950","Text":"At the end, we made a computation and f of x was the area."},{"Start":"03:27.950 ","End":"03:30.035","Text":"We actually wrote the formula,"},{"Start":"03:30.035 ","End":"03:31.670","Text":"y was over there,"},{"Start":"03:31.670 ","End":"03:33.820","Text":"or f of x is x squared plus 2x."},{"Start":"03:33.820 ","End":"03:35.915","Text":"You might be tempted to say,"},{"Start":"03:35.915 ","End":"03:38.165","Text":"when I\u0027m looking at the domain of definition,"},{"Start":"03:38.165 ","End":"03:41.270","Text":"that you are allowed to substitute every value of x."},{"Start":"03:41.270 ","End":"03:44.630","Text":"There is nothing cut out of bounds, when any number,"},{"Start":"03:44.630 ","End":"03:46.470","Text":"positive or negative or whatever,"},{"Start":"03:46.470 ","End":"03:49.880","Text":"we certainly we can square it and take twice of it and add."},{"Start":"03:49.880 ","End":"03:52.145","Text":"But if you wrote all x,"},{"Start":"03:52.145 ","End":"03:53.390","Text":"you would be wrong."},{"Start":"03:53.390 ","End":"03:57.470","Text":"Because this is not a purely mathematical abstract problem."},{"Start":"03:57.470 ","End":"03:59.390","Text":"This comes from the real world."},{"Start":"03:59.390 ","End":"04:02.480","Text":"Here x represents the length of the base of the house."},{"Start":"04:02.480 ","End":"04:05.990","Text":"If x is a length of some part of the house,"},{"Start":"04:05.990 ","End":"04:07.775","Text":"then x can\u0027t be everything."},{"Start":"04:07.775 ","End":"04:10.175","Text":"X has to be strictly positive."},{"Start":"04:10.175 ","End":"04:12.335","Text":"The answer is 2 here, no."},{"Start":"04:12.335 ","End":"04:20.045","Text":"The answer is that x is bigger than 0 because x isn\u0027t abstract,"},{"Start":"04:20.045 ","End":"04:23.180","Text":"it represents a length."},{"Start":"04:23.180 ","End":"04:28.310","Text":"Mathematically there is no restriction that because of the nature of the variable,"},{"Start":"04:28.310 ","End":"04:29.870","Text":"it may be a length,"},{"Start":"04:29.870 ","End":"04:32.990","Text":"may be a weight, which case may be zeros allowed."},{"Start":"04:32.990 ","End":"04:37.820","Text":"Or it could even be a temperature which allows plus or minus."},{"Start":"04:37.820 ","End":"04:40.040","Text":"But if it\u0027s a temperature in centigrade,"},{"Start":"04:40.040 ","End":"04:42.860","Text":"it has to be bigger than minus 273,"},{"Start":"04:42.860 ","End":"04:44.810","Text":"for example, which is absolute 0."},{"Start":"04:44.810 ","End":"04:48.875","Text":"In other words, when you have a problem which represents the situation in the real world,"},{"Start":"04:48.875 ","End":"04:51.170","Text":"you have not only the mathematical limitation,"},{"Start":"04:51.170 ","End":"04:53.600","Text":"but you have the real world limitation which"},{"Start":"04:53.600 ","End":"04:57.300","Text":"determines the domain of definition of a function."}],"ID":8381}],"Thumbnail":null,"ID":1173},{"Name":"The Domain of Basic Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"46s","ChapterTopicVideoID":1121,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/1121.jpeg","UploadDate":"2014-12-25T07:59:49.2330000","DurationForVideoObject":"PT46S","Description":null,"MetaTitle":"Exercise 1 - The Domain of Basic Functions: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on The Domain of Basic Functions practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/function-characteristics/the-domain-of-basic-functions/vid1122","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to find the domain of the function"},{"Start":"00:03.600 ","End":"00:07.650","Text":"y equals x_3 minus x_2 minus 4x plus 1."},{"Start":"00:07.650 ","End":"00:11.415","Text":"The domain simply means the set of values of x,"},{"Start":"00:11.415 ","End":"00:13.140","Text":"which we\u0027re allowed to substitute."},{"Start":"00:13.140 ","End":"00:15.035","Text":"Now, in this case,"},{"Start":"00:15.035 ","End":"00:18.230","Text":"we have what is called a polynomial function,"},{"Start":"00:18.230 ","End":"00:20.495","Text":"and for those of you who don\u0027t remember,"},{"Start":"00:20.495 ","End":"00:29.915","Text":"a polynomial is a function of the form a plus bx plus cx_2 plus etc,"},{"Start":"00:29.915 ","End":"00:35.510","Text":"up to some dx^ n. From the theory,"},{"Start":"00:35.510 ","End":"00:39.485","Text":"you might remember that a polynomial function is defined for all x."},{"Start":"00:39.485 ","End":"00:46.660","Text":"So in this case, we simply write the domain is all x, and we\u0027re done."}],"ID":1122},{"Watched":false,"Name":"Exercise 2","Duration":"49s","ChapterTopicVideoID":1122,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have to find the domain of the function,"},{"Start":"00:03.120 ","End":"00:06.900","Text":"y equals 4x plus 1 over x squared plus 1."},{"Start":"00:06.900 ","End":"00:08.655","Text":"The domain, as you recall,"},{"Start":"00:08.655 ","End":"00:12.615","Text":"means a set of values of x which we\u0027re allowed to substitute."},{"Start":"00:12.615 ","End":"00:18.390","Text":"The only concern here and something that could go wrong would be a denominator of 0."},{"Start":"00:18.390 ","End":"00:20.235","Text":"We have to ask ourselves ,"},{"Start":"00:20.235 ","End":"00:25.095","Text":"which values of x would make the denominator 0 and exclude those?"},{"Start":"00:25.095 ","End":"00:27.615","Text":"Let\u0027s take a look then at the equation,"},{"Start":"00:27.615 ","End":"00:30.750","Text":"x squared plus 1 equals 0."},{"Start":"00:30.750 ","End":"00:35.530","Text":"This would give x squared equals minus 1."},{"Start":"00:35.530 ","End":"00:38.490","Text":"Now the square of a number can never be negative,"},{"Start":"00:38.490 ","End":"00:41.030","Text":"it could be 0 or positive, but never negative."},{"Start":"00:41.030 ","End":"00:49.950","Text":"There\u0027s no such x and so the answer for the domain of the function is all x. We\u0027re done."}],"ID":1123},{"Watched":false,"Name":"Exercise 3","Duration":"1m 7s","ChapterTopicVideoID":1123,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this exercise, we have to find the domain of"},{"Start":"00:02.340 ","End":"00:05.880","Text":"the function y equals 1 over x squared minus 4."},{"Start":"00:05.880 ","End":"00:07.635","Text":"The domain, as you recall,"},{"Start":"00:07.635 ","End":"00:11.910","Text":"simply means the set of values of x which we are allowed to substitute in the function."},{"Start":"00:11.910 ","End":"00:17.190","Text":"The only possible x we couldn\u0027t substitute is 1 which makes the denominator 0."},{"Start":"00:17.190 ","End":"00:19.275","Text":"What we have to make sure, in other words,"},{"Start":"00:19.275 ","End":"00:24.690","Text":"is that x squared minus 4 is not equal to 0."},{"Start":"00:24.690 ","End":"00:28.290","Text":"I suggest that we try and find the values of x for which"},{"Start":"00:28.290 ","End":"00:31.950","Text":"x squared minus 4 is equal to 0,"},{"Start":"00:31.950 ","End":"00:34.290","Text":"and then we\u0027ll exclude those values."},{"Start":"00:34.290 ","End":"00:36.600","Text":"This is a simple quadratic equation."},{"Start":"00:36.600 ","End":"00:42.000","Text":"This can be written as x squared equals 4 and if we take the square root,"},{"Start":"00:42.000 ","End":"00:45.340","Text":"x will be equal to plus or minus the square root of 4,"},{"Start":"00:45.340 ","End":"00:48.170","Text":"in other words, x equals plus or minus 2."},{"Start":"00:48.170 ","End":"00:52.610","Text":"Now, this is a solution to the equation x squared minus 4 equals 0."},{"Start":"00:52.610 ","End":"00:55.485","Text":"In our case, we have to exclude it,"},{"Start":"00:55.485 ","End":"01:00.965","Text":"and we say x is not equal to 2 and x"},{"Start":"01:00.965 ","End":"01:07.860","Text":"is not equal to minus 2 and that\u0027s our domain, and we\u0027re done."}],"ID":1124},{"Watched":false,"Name":"Exercise 4","Duration":"1m 58s","ChapterTopicVideoID":1124,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this exercise, we have to find the domain of"},{"Start":"00:02.640 ","End":"00:06.375","Text":"the function y equals 1 over x cubed minus x."},{"Start":"00:06.375 ","End":"00:09.480","Text":"The domain is simply the set of values of x,"},{"Start":"00:09.480 ","End":"00:11.850","Text":"which we are allowed to substitute in the function."},{"Start":"00:11.850 ","End":"00:14.880","Text":"Now the only thing that could possibly go wrong that we"},{"Start":"00:14.880 ","End":"00:18.600","Text":"couldn\u0027t substitute x would be 1 which makes the denominator 0."},{"Start":"00:18.600 ","End":"00:20.985","Text":"Our condition is basically,"},{"Start":"00:20.985 ","End":"00:26.685","Text":"that x cubed minus x is not equal to 0."},{"Start":"00:26.685 ","End":"00:28.605","Text":"This is an inequality."},{"Start":"00:28.605 ","End":"00:36.840","Text":"The best approach is to try and solve the equation x cubed minus x equals 0,"},{"Start":"00:36.840 ","End":"00:41.550","Text":"and then remember to exclude all the answers to this equation."},{"Start":"00:41.550 ","End":"00:44.945","Text":"Easiest way to go about this is to factor it."},{"Start":"00:44.945 ","End":"00:47.060","Text":"We can take x as a factor out,"},{"Start":"00:47.060 ","End":"00:52.115","Text":"so we get x, x squared minus 1 equals 0."},{"Start":"00:52.115 ","End":"00:55.080","Text":"Now, if a product of 2 numbers is 0,"},{"Start":"00:55.080 ","End":"00:57.450","Text":"then 1 of them has to be 0."},{"Start":"00:57.450 ","End":"01:06.120","Text":"Either x equals 0 or x squared minus 1 equals 0."},{"Start":"01:06.120 ","End":"01:08.775","Text":"If x squared minus 1 is 0,"},{"Start":"01:08.775 ","End":"01:12.255","Text":"then x squared is equal to 1,"},{"Start":"01:12.255 ","End":"01:17.330","Text":"and that gives us that x equals plus or minus 1."},{"Start":"01:17.330 ","End":"01:20.455","Text":"In other words, the solution to the equation,"},{"Start":"01:20.455 ","End":"01:23.300","Text":"x cubed minus 1 equals 0,"},{"Start":"01:23.300 ","End":"01:29.510","Text":"is that x equals 0 or x equals 1,"},{"Start":"01:29.510 ","End":"01:33.535","Text":"or x equals minus 1."},{"Start":"01:33.535 ","End":"01:36.890","Text":"In our case, going back up here,"},{"Start":"01:36.890 ","End":"01:39.815","Text":"if x cubed minus x is not equal to 0,"},{"Start":"01:39.815 ","End":"01:46.520","Text":"we have to exclude these 3 values and conclude that x is not equal to 0,"},{"Start":"01:46.520 ","End":"01:50.495","Text":"and x is not equal to 1,"},{"Start":"01:50.495 ","End":"01:55.160","Text":"and x is not equal to minus 1,"},{"Start":"01:55.160 ","End":"01:58.750","Text":"and that\u0027s our answer. We\u0027re done."}],"ID":1125},{"Watched":false,"Name":"Exercise 5","Duration":"1m 57s","ChapterTopicVideoID":1125,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this exercise, we have to find the domain of the function y"},{"Start":"00:03.150 ","End":"00:06.825","Text":"equals x squared over x squared minus x minus 2."},{"Start":"00:06.825 ","End":"00:11.685","Text":"The domain is a set of values of x which we\u0027re allowed to substitute in the function."},{"Start":"00:11.685 ","End":"00:13.680","Text":"The only thing we might not be able to"},{"Start":"00:13.680 ","End":"00:16.770","Text":"substitute is something which makes the denominator 0."},{"Start":"00:16.770 ","End":"00:20.790","Text":"The domain is described by the inequality,"},{"Start":"00:20.790 ","End":"00:27.285","Text":"x squared minus x minus 2 is not equal to 0."},{"Start":"00:27.285 ","End":"00:28.935","Text":"To solve the inequality,"},{"Start":"00:28.935 ","End":"00:30.960","Text":"we first solve the equation;"},{"Start":"00:30.960 ","End":"00:36.225","Text":"x squared minus x minus 2 is equal to 0."},{"Start":"00:36.225 ","End":"00:38.220","Text":"When get the answer to this,"},{"Start":"00:38.220 ","End":"00:41.170","Text":"these will be the values of x which we have to exclude."},{"Start":"00:41.170 ","End":"00:43.265","Text":"We\u0027re using the quadratic formula."},{"Start":"00:43.265 ","End":"00:45.965","Text":"We get x equals minus b,"},{"Start":"00:45.965 ","End":"00:51.440","Text":"which in this case is 1 plus or minus the square root of b squared,"},{"Start":"00:51.440 ","End":"00:54.095","Text":"is 1 squared minus 4ac,"},{"Start":"00:54.095 ","End":"00:58.740","Text":"which is plus 4 times 1 times 2,"},{"Start":"00:58.740 ","End":"01:00.975","Text":"the plus because we had minus 2,"},{"Start":"01:00.975 ","End":"01:03.590","Text":"all over 2a which is 2."},{"Start":"01:03.590 ","End":"01:11.255","Text":"This is equal to 1 plus or minus the square root of 9 over 2."},{"Start":"01:11.255 ","End":"01:12.980","Text":"If we take the plus,"},{"Start":"01:12.980 ","End":"01:17.715","Text":"we get 1 plus 3 over 2,"},{"Start":"01:17.715 ","End":"01:19.815","Text":"square root of 9 is 3, of course."},{"Start":"01:19.815 ","End":"01:21.435","Text":"If we take the minus,"},{"Start":"01:21.435 ","End":"01:24.855","Text":"we get 1 minus 3 over 2."},{"Start":"01:24.855 ","End":"01:26.925","Text":"In this case we get 2,"},{"Start":"01:26.925 ","End":"01:29.430","Text":"in this case we get minus 1."},{"Start":"01:29.430 ","End":"01:38.465","Text":"To summarize, solution to the equation is x equals 2 or x equals minus 1."},{"Start":"01:38.465 ","End":"01:40.024","Text":"Now if you remember,"},{"Start":"01:40.024 ","End":"01:41.945","Text":"we had an inequality."},{"Start":"01:41.945 ","End":"01:44.090","Text":"If we take these values,"},{"Start":"01:44.090 ","End":"01:47.390","Text":"and in this context of the original inequality,"},{"Start":"01:47.390 ","End":"01:55.715","Text":"we have to say that x is not equal to 2 and x is not equal to minus 1."},{"Start":"01:55.715 ","End":"01:58.410","Text":"That\u0027s our answer. We\u0027re done."}],"ID":1126},{"Watched":false,"Name":"Exercise 6","Duration":"35s","ChapterTopicVideoID":1126,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this exercise, we have to find the domain of"},{"Start":"00:02.670 ","End":"00:06.210","Text":"the function y equals the square root of x minus 4."},{"Start":"00:06.210 ","End":"00:10.995","Text":"The domain is simply the values of x which we\u0027re allowed to substitute in the function."},{"Start":"00:10.995 ","End":"00:13.305","Text":"Notice that we have a square root here."},{"Start":"00:13.305 ","End":"00:16.155","Text":"The requirement is that under the square root sign,"},{"Start":"00:16.155 ","End":"00:17.895","Text":"there shouldn\u0027t be anything negative."},{"Start":"00:17.895 ","End":"00:22.950","Text":"In other words, what we have to have is that x minus 4 must not be negative."},{"Start":"00:22.950 ","End":"00:25.710","Text":"In other words, it must be bigger or equal to 0."},{"Start":"00:25.710 ","End":"00:29.309","Text":"A simple inequality, adding 4 to both sides,"},{"Start":"00:29.309 ","End":"00:32.730","Text":"we get x bigger or equal to 4,"},{"Start":"00:32.730 ","End":"00:35.710","Text":"and that\u0027s our domain. We\u0027re done."}],"ID":1127},{"Watched":false,"Name":"Exercise 7","Duration":"2m 49s","ChapterTopicVideoID":1127,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this exercise, we have to find the domain of the function y"},{"Start":"00:03.240 ","End":"00:07.140","Text":"equals the square root of x squared plus x minus 2."},{"Start":"00:07.140 ","End":"00:12.254","Text":"Now, the domain means the set of all x that can be substituted into the function."},{"Start":"00:12.254 ","End":"00:14.370","Text":"We notice that here we have a square root."},{"Start":"00:14.370 ","End":"00:18.630","Text":"What we have to make sure is that what\u0027s under the square root sign is not negative."},{"Start":"00:18.630 ","End":"00:21.440","Text":"In other words, it has to be bigger or equal to 0."},{"Start":"00:21.440 ","End":"00:25.500","Text":"Basically the domain is defined by the inequality x"},{"Start":"00:25.500 ","End":"00:31.305","Text":"squared plus x minus 2 is bigger or equal to 0."},{"Start":"00:31.305 ","End":"00:32.955","Text":"This is an inequality,"},{"Start":"00:32.955 ","End":"00:34.265","Text":"and to help us solve it,"},{"Start":"00:34.265 ","End":"00:36.650","Text":"we first solve the corresponding equation,"},{"Start":"00:36.650 ","End":"00:42.755","Text":"the equality, x squared plus x minus 2 equals 0."},{"Start":"00:42.755 ","End":"00:45.500","Text":"To solve this, we use the formula,"},{"Start":"00:45.500 ","End":"00:49.935","Text":"and we get x equals minus 1, minus b,"},{"Start":"00:49.935 ","End":"00:54.665","Text":"plus or minus the square root of b squared minus 4ac,"},{"Start":"00:54.665 ","End":"01:01.125","Text":"which is 1 squared minus 4 times 1 times minus 2,"},{"Start":"01:01.125 ","End":"01:03.965","Text":"all over 2a, which is 2."},{"Start":"01:03.965 ","End":"01:10.820","Text":"This is equal to minus 1 plus or minus the square root of 9 over 2."},{"Start":"01:10.820 ","End":"01:13.360","Text":"Taking the plus and the minus,"},{"Start":"01:13.360 ","End":"01:15.180","Text":"square root of 9 is 3, of course."},{"Start":"01:15.180 ","End":"01:19.055","Text":"So we have minus 1 plus 3 over 2,"},{"Start":"01:19.055 ","End":"01:23.015","Text":"and minus 1 minus 3 over 2."},{"Start":"01:23.015 ","End":"01:25.340","Text":"This gives us 1, and the other one,"},{"Start":"01:25.340 ","End":"01:28.440","Text":"minus 4 over 2 is minus 2."},{"Start":"01:28.440 ","End":"01:35.085","Text":"So x is equal to 1 or minus 2,"},{"Start":"01:35.085 ","End":"01:37.520","Text":"these are the two solutions to the equation."},{"Start":"01:37.520 ","End":"01:40.700","Text":"Now, how does this help us with the inequality?"},{"Start":"01:40.700 ","End":"01:43.370","Text":"What we\u0027re going to do is draw a sketch and"},{"Start":"01:43.370 ","End":"01:47.270","Text":"the function x squared plus x minus 2 is a quadratic function,"},{"Start":"01:47.270 ","End":"01:48.695","Text":"which is a parabola."},{"Start":"01:48.695 ","End":"01:51.470","Text":"Because the leading coefficient, in other words,"},{"Start":"01:51.470 ","End":"01:53.180","Text":"a, is bigger than 0,"},{"Start":"01:53.180 ","End":"01:55.280","Text":"it\u0027s going to be an upward parabola."},{"Start":"01:55.280 ","End":"01:57.350","Text":"Let\u0027s make a quick sketch."},{"Start":"01:57.350 ","End":"02:00.200","Text":"We draw the two solutions to the equation,"},{"Start":"02:00.200 ","End":"02:01.340","Text":"1 and minus 2."},{"Start":"02:01.340 ","End":"02:03.280","Text":"Let\u0027s say that 1 is here,"},{"Start":"02:03.280 ","End":"02:05.760","Text":"and maybe minus 2 is here."},{"Start":"02:05.760 ","End":"02:07.715","Text":"Because it\u0027s an upward parabola,"},{"Start":"02:07.715 ","End":"02:10.175","Text":"it will look something like this."},{"Start":"02:10.175 ","End":"02:13.850","Text":"Now, since we want bigger or equal to 0,"},{"Start":"02:13.850 ","End":"02:15.640","Text":"that means above the x-axis,"},{"Start":"02:15.640 ","End":"02:19.150","Text":"so that\u0027s this part and this part,"},{"Start":"02:19.150 ","End":"02:23.730","Text":"and bigger or equal means it includes the points themselves."},{"Start":"02:23.730 ","End":"02:28.820","Text":"From here, we get x bigger or equal to 1,"},{"Start":"02:28.820 ","End":"02:35.625","Text":"and this part or branch is x less than or equal to minus 2."},{"Start":"02:35.625 ","End":"02:37.895","Text":"Altogether when we write the domain,"},{"Start":"02:37.895 ","End":"02:43.185","Text":"we write the domain as x less than or equal to minus 2,"},{"Start":"02:43.185 ","End":"02:47.325","Text":"or x is bigger or equal to 1."},{"Start":"02:47.325 ","End":"02:50.320","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1128},{"Watched":false,"Name":"Exercise 8","Duration":"53s","ChapterTopicVideoID":1128,"CourseChapterTopicPlaylistID":1174,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have to define the domain of the function y"},{"Start":"00:03.720 ","End":"00:08.235","Text":"equals the cube root of x squared plus x minus 1."},{"Start":"00:08.235 ","End":"00:10.500","Text":"The domain being the set of values of"},{"Start":"00:10.500 ","End":"00:13.030","Text":"x which we\u0027re allowed to substitute in the function."},{"Start":"00:13.030 ","End":"00:15.304","Text":"Now in some ways this is a trick question"},{"Start":"00:15.304 ","End":"00:17.855","Text":"because we have a cube root and not a square root."},{"Start":"00:17.855 ","End":"00:20.315","Text":"Now if it was a square root of something,"},{"Start":"00:20.315 ","End":"00:23.525","Text":"we\u0027d have to make sure that what\u0027s under the square root is not negative."},{"Start":"00:23.525 ","End":"00:26.705","Text":"For example, the square root of minus 4 does not exist."},{"Start":"00:26.705 ","End":"00:28.850","Text":"But it\u0027s very important that here we have"},{"Start":"00:28.850 ","End":"00:31.700","Text":"a cube root and there are no problems with cube roots."},{"Start":"00:31.700 ","End":"00:34.190","Text":"If we have a cube root of say,"},{"Start":"00:34.190 ","End":"00:36.900","Text":"8, a positive number, it\u0027s 2,"},{"Start":"00:36.900 ","End":"00:42.560","Text":"and the cube root of a negative number like minus 8 is also defined minus 2,"},{"Start":"00:42.560 ","End":"00:46.385","Text":"so it makes no difference positive or negative, everything works."},{"Start":"00:46.385 ","End":"00:53.520","Text":"In brief, the domain of the function is simply all x, and we\u0027re done."}],"ID":1129}],"Thumbnail":null,"ID":1174},{"Name":"The Domain of Logarithmic and Exponential Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"3m 51s","ChapterTopicVideoID":31847,"CourseChapterTopicPlaylistID":1175,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34099},{"Watched":false,"Name":"Exercise 2","Duration":"3m 14s","ChapterTopicVideoID":31845,"CourseChapterTopicPlaylistID":1175,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34100},{"Watched":false,"Name":"Exercise 3","Duration":"3m 35s","ChapterTopicVideoID":31846,"CourseChapterTopicPlaylistID":1175,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34101},{"Watched":false,"Name":"Exercise 4","Duration":"1m 42s","ChapterTopicVideoID":31844,"CourseChapterTopicPlaylistID":1175,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34102},{"Watched":false,"Name":"Exercise 5","Duration":"6m 23s","ChapterTopicVideoID":31848,"CourseChapterTopicPlaylistID":1175,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34103}],"Thumbnail":null,"ID":1175},{"Name":"The Domain of Trigonometric Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"35s","ChapterTopicVideoID":3321,"CourseChapterTopicPlaylistID":1176,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"In this exercise, we have to find the domain of the function"},{"Start":"00:03.285 ","End":"00:07.665","Text":"y equals sine 2x plus cosine 3x."},{"Start":"00:07.665 ","End":"00:12.764","Text":"The domain, meaning the values of x that we\u0027re allowed to substitute in the function."},{"Start":"00:12.764 ","End":"00:15.030","Text":"Now, this function is made up of 2 parts,"},{"Start":"00:15.030 ","End":"00:16.515","Text":"sine and a cosine."},{"Start":"00:16.515 ","End":"00:19.890","Text":"The sine is defined for all x."},{"Start":"00:19.890 ","End":"00:22.950","Text":"The cosine is defined for all x and"},{"Start":"00:22.950 ","End":"00:27.240","Text":"certainly for any x they have no problem in computing 2x or 3x."},{"Start":"00:27.240 ","End":"00:29.850","Text":"There are really no restrictions on x here"},{"Start":"00:29.850 ","End":"00:33.450","Text":"and the answer is simply all x."},{"Start":"00:33.450 ","End":"00:35.470","Text":"That\u0027s it."}],"ID":3332},{"Watched":false,"Name":"Exercise 2","Duration":"1m 53s","ChapterTopicVideoID":3322,"CourseChapterTopicPlaylistID":1176,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.510","Text":"In this exercise, we have to find the domain of the function y equals tangent of 2x."},{"Start":"00:06.510 ","End":"00:10.890","Text":"Remember, the domain simply means the values of x we\u0027re allowed to substitute."},{"Start":"00:10.890 ","End":"00:12.885","Text":"Now, to make life easier,"},{"Start":"00:12.885 ","End":"00:15.840","Text":"we\u0027ll write the tangent in terms of sine and cosine."},{"Start":"00:15.840 ","End":"00:25.060","Text":"A function can be rewritten as y equals sine 2x over cosine of 2x."},{"Start":"00:25.060 ","End":"00:31.035","Text":"Now the domain of sine is all x and the domain of cosine is also all x,"},{"Start":"00:31.035 ","End":"00:33.375","Text":"so what might cause a problem?"},{"Start":"00:33.375 ","End":"00:37.130","Text":"The only thing that\u0027s not good is that the denominator could be 0."},{"Start":"00:37.130 ","End":"00:38.985","Text":"That\u0027s the only restriction."},{"Start":"00:38.985 ","End":"00:44.644","Text":"We have to ask that cosine 2x not be equal to 0."},{"Start":"00:44.644 ","End":"00:51.065","Text":"Now, remember that cosine for some angle Alpha is equal to 0,"},{"Start":"00:51.065 ","End":"01:00.100","Text":"has a general solution that Alpha is equal to Pi over 2 plus k times Pi,"},{"Start":"01:00.100 ","End":"01:03.245","Text":"where k is any integer."},{"Start":"01:03.245 ","End":"01:07.175","Text":"In our case, we put Alpha equals 2x."},{"Start":"01:07.175 ","End":"01:10.745","Text":"Cosine 2x is equal to 0,"},{"Start":"01:10.745 ","End":"01:12.160","Text":"which is what we don\u0027t want."},{"Start":"01:12.160 ","End":"01:18.945","Text":"When 2x is equal to Pi over 2 plus k Pi,"},{"Start":"01:18.945 ","End":"01:22.245","Text":"which I\u0027m just taking from here, dividing by 2,"},{"Start":"01:22.245 ","End":"01:29.865","Text":"I get x equals Pi over 4 plus k times Pi over 2,"},{"Start":"01:29.865 ","End":"01:31.955","Text":"where k is an integer,"},{"Start":"01:31.955 ","End":"01:37.040","Text":"sometimes written as k belongs to the set of integers."},{"Start":"01:37.040 ","End":"01:39.140","Text":"Of course, that\u0027s when the denominator is 0,"},{"Start":"01:39.140 ","End":"01:41.540","Text":"so our actual answer would be,"},{"Start":"01:41.540 ","End":"01:45.245","Text":"the domain is that x must not equal to"},{"Start":"01:45.245 ","End":"01:53.550","Text":"Pi over 4 plus k times Pi over 2. We\u0027re done."}],"ID":3333},{"Watched":false,"Name":"Exercise 3","Duration":"1m 47s","ChapterTopicVideoID":3332,"CourseChapterTopicPlaylistID":1176,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.465","Text":"In this exercise, we have to find the domain of the function y equals cotangent to 4x."},{"Start":"00:06.465 ","End":"00:11.460","Text":"Remember, the domain simply means the values of x that we\u0027re allowed to substitute."},{"Start":"00:11.460 ","End":"00:14.610","Text":"Now, when we see a trigonometric function like cotangent,"},{"Start":"00:14.610 ","End":"00:17.460","Text":"it\u0027s easier to write it in terms of sine and cosine."},{"Start":"00:17.460 ","End":"00:26.405","Text":"Let\u0027s rewrite our function as y equals cosine of 4x over sine 4x."},{"Start":"00:26.405 ","End":"00:30.535","Text":"Remembering of course that cotangent is cosine over sine."},{"Start":"00:30.535 ","End":"00:36.625","Text":"Now, the domain of the cosine is all x and the domain of the sine is all x."},{"Start":"00:36.625 ","End":"00:40.355","Text":"What might the problem be for sum x?"},{"Start":"00:40.355 ","End":"00:43.460","Text":"The only thing bad that could be is that the denominator"},{"Start":"00:43.460 ","End":"00:46.745","Text":"could turn out to be 0 and this is what we don\u0027t want."},{"Start":"00:46.745 ","End":"00:53.840","Text":"In other words, we have to insist that sine of 4x is not equal to 0."},{"Start":"00:53.840 ","End":"01:00.980","Text":"Now, just a reminder that sine of an angle Alpha is equal to 0 has"},{"Start":"01:00.980 ","End":"01:08.720","Text":"the solution that Alpha is equal to some integer k times Pi,"},{"Start":"01:08.720 ","End":"01:10.219","Text":"k is any integer."},{"Start":"01:10.219 ","End":"01:14.305","Text":"In our case, instead of Alpha, we have 4x."},{"Start":"01:14.305 ","End":"01:16.425","Text":"Getting back to our case,"},{"Start":"01:16.425 ","End":"01:19.665","Text":"sine of 4x is equal to 0,"},{"Start":"01:19.665 ","End":"01:21.690","Text":"and this is what we\u0027re going to exclude later,"},{"Start":"01:21.690 ","End":"01:25.800","Text":"when 4x is equal to k Pi."},{"Start":"01:25.800 ","End":"01:32.255","Text":"In other words, x equals k times Pi over 4."},{"Start":"01:32.255 ","End":"01:35.315","Text":"These are the values which make the denominator 0,"},{"Start":"01:35.315 ","End":"01:40.280","Text":"so our domain is simply x not equal to"},{"Start":"01:40.280 ","End":"01:47.940","Text":"k times Pi over 4 for any integer k and that\u0027s our answer."}],"ID":3343},{"Watched":false,"Name":"Exercise 4","Duration":"55s","ChapterTopicVideoID":3333,"CourseChapterTopicPlaylistID":1176,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.940","Text":"In this exercise, we have to define the domain of the function y equals arc sine of 2x."},{"Start":"00:05.940 ","End":"00:07.770","Text":"Remember the domain in simple terms,"},{"Start":"00:07.770 ","End":"00:11.185","Text":"just means which values of x are we allowed to substitute here."},{"Start":"00:11.185 ","End":"00:14.730","Text":"We have to remember some basic trigonometry that in general,"},{"Start":"00:14.730 ","End":"00:20.355","Text":"if we have a function arc sine of some variable say a,"},{"Start":"00:20.355 ","End":"00:27.485","Text":"this is defined for all values of a between minus 1 and 1 inclusive."},{"Start":"00:27.485 ","End":"00:30.955","Text":"In our case, our a is 2x."},{"Start":"00:30.955 ","End":"00:35.420","Text":"Basically, what we have to do is just take the general rule and apply it to"},{"Start":"00:35.420 ","End":"00:40.785","Text":"2x so we get minus 1 less than or equal to 2x,"},{"Start":"00:40.785 ","End":"00:42.825","Text":"less than or equal to 1,"},{"Start":"00:42.825 ","End":"00:44.765","Text":"and dividing by 2,"},{"Start":"00:44.765 ","End":"00:52.735","Text":"we get x less than or equal to 1/2 but bigger or equal to minus 1/2."},{"Start":"00:52.735 ","End":"00:56.190","Text":"That\u0027s basically the answer. We\u0027re done."}],"ID":3344},{"Watched":false,"Name":"Exercise 5","Duration":"54s","ChapterTopicVideoID":3334,"CourseChapterTopicPlaylistID":1176,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we have to find the domain of the function y equals arccosine of 3x."},{"Start":"00:06.030 ","End":"00:10.440","Text":"We have to remember some basic trigonometry that if we have a function or"},{"Start":"00:10.440 ","End":"00:14.910","Text":"rather an expression like arccosine of some variable,"},{"Start":"00:14.910 ","End":"00:19.020","Text":"say a, that it\u0027s defined a certain values of a,"},{"Start":"00:19.020 ","End":"00:20.310","Text":"in this particular case,"},{"Start":"00:20.310 ","End":"00:21.750","Text":"between minus 1 and 1."},{"Start":"00:21.750 ","End":"00:25.755","Text":"In other words, minus 1 is less than or equal to a,"},{"Start":"00:25.755 ","End":"00:28.350","Text":"which is less than or equal to 1."},{"Start":"00:28.350 ","End":"00:32.470","Text":"In our case, a is 3x."},{"Start":"00:32.960 ","End":"00:43.305","Text":"Putting a here for 3x minus 1 less than or equal to 3x less than or equal to 1,"},{"Start":"00:43.305 ","End":"00:45.180","Text":"divided by 3, we get,"},{"Start":"00:45.180 ","End":"00:49.279","Text":"minus 1/3 less than or equal to x,"},{"Start":"00:49.279 ","End":"00:51.760","Text":"less than or equal to 1/3,"},{"Start":"00:51.760 ","End":"00:54.820","Text":"and that\u0027s the answer. We\u0027re done."}],"ID":3345}],"Thumbnail":null,"ID":1176},{"Name":"The Domain of Absolute Value Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"2m 38s","ChapterTopicVideoID":4353,"CourseChapterTopicPlaylistID":1177,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have to find the domain of the function y equals"},{"Start":"00:03.510 ","End":"00:07.215","Text":"1 over the square root of absolute value of x minus 4."},{"Start":"00:07.215 ","End":"00:09.660","Text":"Remember that the domain in simple terms means"},{"Start":"00:09.660 ","End":"00:12.360","Text":"which values of x are we allowed to substitute."},{"Start":"00:12.360 ","End":"00:14.020","Text":"We have several elements here,"},{"Start":"00:14.020 ","End":"00:15.645","Text":"we have an absolute value,"},{"Start":"00:15.645 ","End":"00:16.935","Text":"we have a square root,"},{"Start":"00:16.935 ","End":"00:18.675","Text":"and we have a denominator."},{"Start":"00:18.675 ","End":"00:21.660","Text":"Now the absolute value of x has no problems,"},{"Start":"00:21.660 ","End":"00:23.145","Text":"it\u0027s defined for all x."},{"Start":"00:23.145 ","End":"00:25.500","Text":"But there\u0027s 2 things we do have to watch out for."},{"Start":"00:25.500 ","End":"00:30.570","Text":"What\u0027s under the square root has to be non-negative or bigger or equal to 0."},{"Start":"00:30.570 ","End":"00:33.340","Text":"Also we can\u0027t have a 0 in the denominator."},{"Start":"00:33.340 ","End":"00:36.830","Text":"Let\u0027s first of all take the matter of the square root."},{"Start":"00:36.830 ","End":"00:42.200","Text":"What\u0027s under the square root is absolute value of x minus 4,"},{"Start":"00:42.200 ","End":"00:45.319","Text":"and that has to be bigger or equal to 0."},{"Start":"00:45.319 ","End":"00:51.455","Text":"This gives us the absolute value of x is bigger or equal to 4."},{"Start":"00:51.455 ","End":"00:53.690","Text":"This is a familiar type of inequality."},{"Start":"00:53.690 ","End":"00:57.440","Text":"I\u0027ll just remind you that when we have absolute value of x bigger"},{"Start":"00:57.440 ","End":"01:01.355","Text":"or equal to some value a, some positive number,"},{"Start":"01:01.355 ","End":"01:06.410","Text":"then the solution is that x is less than or equal to minus a,"},{"Start":"01:06.410 ","End":"01:10.500","Text":"or x is bigger or equal to a."},{"Start":"01:10.500 ","End":"01:15.680","Text":"In our case we have that x is less than or equal to minus 4,"},{"Start":"01:15.680 ","End":"01:18.830","Text":"or x bigger or equal to 4."},{"Start":"01:18.830 ","End":"01:22.115","Text":"Now, that takes care of the square root."},{"Start":"01:22.115 ","End":"01:24.740","Text":"That\u0027s what\u0027s under the square root is non-negative."},{"Start":"01:24.740 ","End":"01:28.536","Text":"Next thing to do is to take care of the matter of the denominator"},{"Start":"01:28.536 ","End":"01:30.200","Text":"that it shouldn\u0027t be 0."},{"Start":"01:30.200 ","End":"01:34.520","Text":"The denominator is the square root of something."},{"Start":"01:34.520 ","End":"01:40.123","Text":"We have to say that the square root of absolute value of x minus 4"},{"Start":"01:40.123 ","End":"01:42.800","Text":"is not equal to 0."},{"Start":"01:42.800 ","End":"01:45.170","Text":"Now, the square root of something is not 0,"},{"Start":"01:45.170 ","End":"01:47.435","Text":"the thing itself is not 0."},{"Start":"01:47.435 ","End":"01:53.475","Text":"Absolute value of x minus 4 is not equal to 0."},{"Start":"01:53.475 ","End":"01:55.260","Text":"Take the 4 over to the other side,"},{"Start":"01:55.260 ","End":"01:59.380","Text":"absolute value of x is not equal to 4."},{"Start":"01:59.380 ","End":"02:05.975","Text":"That means that x is not equal to minus 4 or 4."},{"Start":"02:05.975 ","End":"02:09.170","Text":"Now, if we combine what we have here,"},{"Start":"02:09.170 ","End":"02:12.440","Text":"which is the condition for the denominator not being"},{"Start":"02:12.440 ","End":"02:16.580","Text":"0 and this condition which is related to the square root"},{"Start":"02:16.580 ","End":"02:18.440","Text":"and combine them together,"},{"Start":"02:18.440 ","End":"02:22.235","Text":"what we\u0027ll get is that the equals is not allowed."},{"Start":"02:22.235 ","End":"02:25.920","Text":"We summarize it by saying that x is less"},{"Start":"02:25.920 ","End":"02:30.245","Text":"than minus 4 instead of less than or equal to because of this."},{"Start":"02:30.245 ","End":"02:35.470","Text":"Or x is strictly bigger than 4."},{"Start":"02:35.470 ","End":"02:38.770","Text":"That\u0027s our domain and we\u0027re done."}],"ID":4362},{"Watched":false,"Name":"Exercise 2","Duration":"58s","ChapterTopicVideoID":4836,"CourseChapterTopicPlaylistID":1177,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.400","Text":"In this exercise, we have to find the domain of the function as follows,"},{"Start":"00:04.400 ","End":"00:08.385","Text":"y equals natural log of 2 minus absolute value of x."},{"Start":"00:08.385 ","End":"00:10.665","Text":"Let\u0027s see where problems might be."},{"Start":"00:10.665 ","End":"00:15.705","Text":"The absolute value of x is certainly defined for all x and we can certainly do 2 minus."},{"Start":"00:15.705 ","End":"00:18.780","Text":"The only problem might come from the natural logarithm."},{"Start":"00:18.780 ","End":"00:23.550","Text":"Remember, the domain of the natural logarithm is the positive numbers."},{"Start":"00:23.550 ","End":"00:25.890","Text":"It\u0027s only defined on strictly positive."},{"Start":"00:25.890 ","End":"00:28.980","Text":"What it means is that the argument of the natural log,"},{"Start":"00:28.980 ","End":"00:31.920","Text":"which is 2 minus the absolute value of x,"},{"Start":"00:31.920 ","End":"00:34.305","Text":"has to be strictly positive."},{"Start":"00:34.305 ","End":"00:38.050","Text":"If we change this inequality around a bit,"},{"Start":"00:38.050 ","End":"00:40.730","Text":"we put this to the other side and switch sides,"},{"Start":"00:40.730 ","End":"00:44.735","Text":"we\u0027ll get the absolute value of x is less than 2."},{"Start":"00:44.735 ","End":"00:47.570","Text":"For the absolute value of x to be less than 2,"},{"Start":"00:47.570 ","End":"00:54.020","Text":"we have to have that x has to be between 2 and minus 2."},{"Start":"00:54.020 ","End":"00:55.805","Text":"That\u0027s all there is to it."},{"Start":"00:55.805 ","End":"00:58.710","Text":"This is the answer and we\u0027re done."}],"ID":4836}],"Thumbnail":null,"ID":1177},{"Name":"The Domain of a Piecewise Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"4m 33s","ChapterTopicVideoID":4837,"CourseChapterTopicPlaylistID":1178,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have to find the domain of the function f of x,"},{"Start":"00:04.650 ","End":"00:07.665","Text":"which is defined piece-wise in 3 separate pieces."},{"Start":"00:07.665 ","End":"00:10.590","Text":"Here, it\u0027s defined this way, and so on."},{"Start":"00:10.590 ","End":"00:13.860","Text":"What we want to do now is to look"},{"Start":"00:13.860 ","End":"00:16.890","Text":"at each piece separately and see if,"},{"Start":"00:16.890 ","End":"00:19.470","Text":"I call it further restrictions apply."},{"Start":"00:19.470 ","End":"00:24.090","Text":"The first piece is x bigger than 0,"},{"Start":"00:24.090 ","End":"00:30.505","Text":"and here we have to see if 1 over x really is defined on all of this,"},{"Start":"00:30.505 ","End":"00:32.070","Text":"and the answer is yes."},{"Start":"00:32.070 ","End":"00:37.050","Text":"No further restrictions apply because 1 over x is only not defined when x is 0,"},{"Start":"00:37.050 ","End":"00:38.655","Text":"which is not part of this,"},{"Start":"00:38.655 ","End":"00:44.715","Text":"so x bigger than 0 stays as is."},{"Start":"00:44.715 ","End":"00:51.450","Text":"The next bit is when x is between 0 and 4, including the 0,"},{"Start":"00:51.450 ","End":"00:53.070","Text":"not including the 4,"},{"Start":"00:53.070 ","End":"00:54.640","Text":"and here we have to look at"},{"Start":"00:54.640 ","End":"00:58.450","Text":"the square root function and see if it\u0027s restricted in any way."},{"Start":"00:58.450 ","End":"01:02.770","Text":"Well, the square root is defined for all non-negative numbers,"},{"Start":"01:02.770 ","End":"01:04.450","Text":"meaning x bigger or equal to 0,"},{"Start":"01:04.450 ","End":"01:06.955","Text":"which is certainly true and all of this."},{"Start":"01:06.955 ","End":"01:09.730","Text":"Again, no further restrictions apply,"},{"Start":"01:09.730 ","End":"01:12.579","Text":"and all of this is part of the domain."},{"Start":"01:12.579 ","End":"01:14.830","Text":"On the third piece,"},{"Start":"01:14.830 ","End":"01:20.790","Text":"we have x between 4 and 20, not inclusive."},{"Start":"01:20.790 ","End":"01:25.440","Text":"Here we have to look at 1 over x minus 8."},{"Start":"01:25.440 ","End":"01:31.470","Text":"Finally, something interesting happens is that this function, in general,"},{"Start":"01:31.470 ","End":"01:35.810","Text":"would not be defined where the denominator would be 0,"},{"Start":"01:35.810 ","End":"01:37.610","Text":"which is x equals 8,"},{"Start":"01:37.610 ","End":"01:44.030","Text":"so what we have to do is take this and remove x equals 8."},{"Start":"01:44.030 ","End":"01:49.500","Text":"This becomes this,"},{"Start":"01:49.500 ","End":"01:51.750","Text":"I\u0027ll write the word but, which means and,"},{"Start":"01:51.750 ","End":"01:56.140","Text":"but x not equal to 8."},{"Start":"01:56.270 ","End":"02:01.890","Text":"This is the third piece, 1, 2, 3."},{"Start":"02:01.890 ","End":"02:03.979","Text":"Now we have to combine them somehow."},{"Start":"02:03.979 ","End":"02:06.320","Text":"I could do it graphically on the number line,"},{"Start":"02:06.320 ","End":"02:08.545","Text":"but I don\u0027t think that\u0027s necessary."},{"Start":"02:08.545 ","End":"02:11.570","Text":"What I want to do is slightly simplify this."},{"Start":"02:11.570 ","End":"02:13.910","Text":"Look, if I take from 4-20 and I remove 8,"},{"Start":"02:13.910 ","End":"02:16.565","Text":"I can write this into 2 separate bits, so the last bit,"},{"Start":"02:16.565 ","End":"02:22.910","Text":"I\u0027m going to rewrite as 4 less than x,"},{"Start":"02:22.910 ","End":"02:31.965","Text":"less than 8, or 8 less than x, less than 20."},{"Start":"02:31.965 ","End":"02:33.320","Text":"If I write them all,"},{"Start":"02:33.320 ","End":"02:37.430","Text":"so what I get is that x is bigger than 0,"},{"Start":"02:37.430 ","End":"02:45.030","Text":"or x is between 0 and 4,"},{"Start":"02:45.030 ","End":"02:47.250","Text":"including the 0, not including the 4,"},{"Start":"02:47.250 ","End":"02:50.475","Text":"or this or this,"},{"Start":"02:50.475 ","End":"02:53.555","Text":"which is customary to simplify a bit."},{"Start":"02:53.555 ","End":"02:58.030","Text":"For example, let\u0027s look at the first 2."},{"Start":"02:58.030 ","End":"03:01.085","Text":"Oh, sorry, I wrote this the wrong way."},{"Start":"03:01.085 ","End":"03:02.690","Text":"Why didn\u0027t you say anything?"},{"Start":"03:02.690 ","End":"03:04.325","Text":"Here we are now."},{"Start":"03:04.325 ","End":"03:08.330","Text":"This and this certainly can combine,"},{"Start":"03:08.330 ","End":"03:09.860","Text":"and I\u0027ll just write the answer,"},{"Start":"03:09.860 ","End":"03:11.240","Text":"and you\u0027ll see that this is so."},{"Start":"03:11.240 ","End":"03:17.670","Text":"These 2 can combine to say that x is less than 4 because,"},{"Start":"03:17.670 ","End":"03:19.690","Text":"look, this is all the negatives up to,"},{"Start":"03:19.690 ","End":"03:21.910","Text":"not including 0, then the 0,"},{"Start":"03:21.910 ","End":"03:27.230","Text":"and up to 4, so altogether we\u0027re continuous on everything up to 4,"},{"Start":"03:27.230 ","End":"03:35.535","Text":"so I would write the answer as x less than 4,"},{"Start":"03:35.535 ","End":"03:41.130","Text":"or x is between 4 and 8,"},{"Start":"03:41.130 ","End":"03:47.505","Text":"or x is between 8 and 20."},{"Start":"03:47.505 ","End":"03:49.380","Text":"That\u0027s 1 way of writing it,"},{"Start":"03:49.380 ","End":"03:52.695","Text":"and I\u0027ll give that as the answer."},{"Start":"03:52.695 ","End":"03:55.680","Text":"In fact, I\u0027ll even highlight it,"},{"Start":"03:55.680 ","End":"03:59.735","Text":"but that\u0027s not the only possible answer."},{"Start":"03:59.735 ","End":"04:01.880","Text":"I\u0027ll show you 1 other possible way you could do it"},{"Start":"04:01.880 ","End":"04:04.400","Text":"if you\u0027re creative or if you think differently."},{"Start":"04:04.400 ","End":"04:07.955","Text":"You could say we have all the x\u0027s up to 20,"},{"Start":"04:07.955 ","End":"04:12.425","Text":"so I could say that x is less than 20,"},{"Start":"04:12.425 ","End":"04:15.615","Text":"but, oftentimes I write it as just a comma,"},{"Start":"04:15.615 ","End":"04:17.580","Text":"2 values are missing."},{"Start":"04:17.580 ","End":"04:18.780","Text":"The 2 wholes,"},{"Start":"04:18.780 ","End":"04:21.045","Text":"x not equal to 4,"},{"Start":"04:21.045 ","End":"04:24.555","Text":"and x not equal to 8."},{"Start":"04:24.555 ","End":"04:26.010","Text":"Really, this comma is an and."},{"Start":"04:26.010 ","End":"04:27.600","Text":"X is less than 20,"},{"Start":"04:27.600 ","End":"04:29.850","Text":"but not 4 or 8."},{"Start":"04:29.850 ","End":"04:32.360","Text":"That\u0027s another possibility that you could,"},{"Start":"04:32.360 ","End":"04:36.030","Text":"both of these are okay. I\u0027m done."}],"ID":4837}],"Thumbnail":null,"ID":1178},{"Name":"Translation (Shifting) and Reflection of Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vertical Translation of Basic Functions","Duration":"3m 11s","ChapterTopicVideoID":8241,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this clip and in the next one,"},{"Start":"00:02.565 ","End":"00:05.489","Text":"we\u0027ll be talking about translation of function."},{"Start":"00:05.489 ","End":"00:07.845","Text":"In this section, it will be vertical,"},{"Start":"00:07.845 ","End":"00:09.825","Text":"and then the next we\u0027ll do the horizontal."},{"Start":"00:09.825 ","End":"00:11.565","Text":"Vertical meaning up and down,"},{"Start":"00:11.565 ","End":"00:13.875","Text":"translation is another way of saying shift."},{"Start":"00:13.875 ","End":"00:16.050","Text":"But perhaps, it\u0027s best to start with an example."},{"Start":"00:16.050 ","End":"00:18.165","Text":"Let\u0027s take any function,"},{"Start":"00:18.165 ","End":"00:20.735","Text":"let\u0027s say y equals x squared."},{"Start":"00:20.735 ","End":"00:24.280","Text":"We\u0027ll take y equals x squared."},{"Start":"00:24.280 ","End":"00:29.940","Text":"Here\u0027s the y equals x squared which I got by simply plotting few points,"},{"Start":"00:29.940 ","End":"00:31.740","Text":"0, 0, 1, 1, 2, 4,"},{"Start":"00:31.740 ","End":"00:33.860","Text":"etc, and joining them with the line."},{"Start":"00:33.860 ","End":"00:35.330","Text":"The point is not to be accurate,"},{"Start":"00:35.330 ","End":"00:37.160","Text":"it\u0027s to get the general idea."},{"Start":"00:37.160 ","End":"00:43.215","Text":"Someone asks me to take this function and move it upwards 1 unit to shift it upwards,"},{"Start":"00:43.215 ","End":"00:45.695","Text":"what we call a vertical translation upwards."},{"Start":"00:45.695 ","End":"00:47.180","Text":"What would it look like?"},{"Start":"00:47.180 ","End":"00:51.500","Text":"Well, I could just take all these points and move them 1 up and I get these points."},{"Start":"00:51.500 ","End":"00:55.295","Text":"Likewise, I could connect them with this continuous line,"},{"Start":"00:55.295 ","End":"01:00.000","Text":"something like this so that if this is where y equals 0,"},{"Start":"01:00.000 ","End":"01:01.785","Text":"here y equals 1."},{"Start":"01:01.785 ","End":"01:04.095","Text":"If here why was 1 here would be 2."},{"Start":"01:04.095 ","End":"01:05.950","Text":"Each time we add 1,"},{"Start":"01:05.950 ","End":"01:09.830","Text":"we move it upwards so that this distance here is 1."},{"Start":"01:09.830 ","End":"01:13.355","Text":"The question is, what is the equation of this new function?"},{"Start":"01:13.355 ","End":"01:16.840","Text":"Well quite clearly, if I moved it up with 1,"},{"Start":"01:16.840 ","End":"01:20.580","Text":"the equation would be y equals x squared plus 1."},{"Start":"01:20.580 ","End":"01:23.935","Text":"Whatever was here was x squared and then we add 1."},{"Start":"01:23.935 ","End":"01:27.265","Text":"At every point, we go upwards by one unit."},{"Start":"01:27.265 ","End":"01:30.110","Text":"That\u0027s upward translation or upward shifting."},{"Start":"01:30.110 ","End":"01:31.670","Text":"Now, what about the reverse direction?"},{"Start":"01:31.670 ","End":"01:34.340","Text":"Now, I\u0027m asked to move the original function,"},{"Start":"01:34.340 ","End":"01:37.515","Text":"y equals x squared down by 2 units."},{"Start":"01:37.515 ","End":"01:39.270","Text":"Again, first of all,"},{"Start":"01:39.270 ","End":"01:40.640","Text":"I plot some points,"},{"Start":"01:40.640 ","End":"01:43.750","Text":"something like this, and then I join them up with the line."},{"Start":"01:43.750 ","End":"01:48.260","Text":"The way we get from the black to the blue is by going 2 units down."},{"Start":"01:48.260 ","End":"01:50.300","Text":"Here I go downwards,"},{"Start":"01:50.300 ","End":"01:54.260","Text":"2 units, downwards 2 units, and so on."},{"Start":"01:54.260 ","End":"01:58.440","Text":"The equation of this graph quite clearly would be"},{"Start":"01:58.440 ","End":"02:04.340","Text":"the original x squared to get to the point on the black and then move it down by 2."},{"Start":"02:04.340 ","End":"02:06.350","Text":"Now I\u0027d like to generalize this concept."},{"Start":"02:06.350 ","End":"02:08.540","Text":"What if it wasn\u0027t x squared?"},{"Start":"02:08.540 ","End":"02:11.100","Text":"Suppose it was just any old function."},{"Start":"02:11.100 ","End":"02:13.250","Text":"If it was just some f(x),"},{"Start":"02:13.250 ","End":"02:14.990","Text":"surely the same principle holds."},{"Start":"02:14.990 ","End":"02:16.340","Text":"There\u0027s nothing special about x squared."},{"Start":"02:16.340 ","End":"02:18.110","Text":"Any function that I take,"},{"Start":"02:18.110 ","End":"02:20.135","Text":"I can raise and lower,"},{"Start":"02:20.135 ","End":"02:22.790","Text":"translate upwards or downwards, shift up,"},{"Start":"02:22.790 ","End":"02:24.500","Text":"shift down many names,"},{"Start":"02:24.500 ","End":"02:26.210","Text":"the same principle would hold."},{"Start":"02:26.210 ","End":"02:29.105","Text":"Now I\u0027d like to take the generalization still further."},{"Start":"02:29.105 ","End":"02:31.520","Text":"What if it wasn\u0027t 1 unit I raised it by?"},{"Start":"02:31.520 ","End":"02:33.800","Text":"What if it wasn\u0027t 2 units I lowered it by?"},{"Start":"02:33.800 ","End":"02:38.585","Text":"Suppose it with some general k upwards or l downwards,"},{"Start":"02:38.585 ","End":"02:41.905","Text":"surely I could replace 1 by k and 2 by l,"},{"Start":"02:41.905 ","End":"02:43.820","Text":"k upwards, l downwards."},{"Start":"02:43.820 ","End":"02:45.180","Text":"This brings us to the point k,"},{"Start":"02:45.180 ","End":"02:48.115","Text":"this is the point minus l because it\u0027s downwards."},{"Start":"02:48.115 ","End":"02:52.430","Text":"As a summary, what I can tell you about the vertical translation of functions,"},{"Start":"02:52.430 ","End":"02:57.770","Text":"is that if you take a function that you know and you want to shift it up by k units,"},{"Start":"02:57.770 ","End":"03:03.290","Text":"you just add plus k. If you want to shift some function fx downwards by l units,"},{"Start":"03:03.290 ","End":"03:05.550","Text":"you just write minus l after the function,"},{"Start":"03:05.550 ","End":"03:07.010","Text":"and that gives you the new function."},{"Start":"03:07.010 ","End":"03:08.390","Text":"We\u0027re done with vertical."},{"Start":"03:08.390 ","End":"03:12.000","Text":"Now let\u0027s move on to the horizontal, left and right."}],"ID":8401},{"Watched":false,"Name":"Horizontal Translation of Basic Functions","Duration":"4m 19s","ChapterTopicVideoID":8243,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.795","Text":"Here we are once again with translation of functions,"},{"Start":"00:03.795 ","End":"00:06.510","Text":"and you can remember that translation means shifting,"},{"Start":"00:06.510 ","End":"00:09.600","Text":"only this time it\u0027ll be horizontal rather than vertical."},{"Start":"00:09.600 ","End":"00:10.695","Text":"By which I mean,"},{"Start":"00:10.695 ","End":"00:12.655","Text":"we\u0027ll be shifting left and right,"},{"Start":"00:12.655 ","End":"00:15.330","Text":"whereas previously we were shifting up and down."},{"Start":"00:15.330 ","End":"00:17.820","Text":"It\u0027s a bit different as you will soon see."},{"Start":"00:17.820 ","End":"00:20.190","Text":"Once again, let\u0027s take some function."},{"Start":"00:20.190 ","End":"00:23.385","Text":"They might as well be y equals x squared."},{"Start":"00:23.385 ","End":"00:26.760","Text":"Someone asks me, would I please move it"},{"Start":"00:26.760 ","End":"00:30.690","Text":"4 units to the right and tell me what the new function will be?"},{"Start":"00:30.690 ","End":"00:32.100","Text":"As before, what I would do,"},{"Start":"00:32.100 ","End":"00:33.780","Text":"I would take these points that are used"},{"Start":"00:33.780 ","End":"00:35.250","Text":"in the sketching and move them"},{"Start":"00:35.250 ","End":"00:38.850","Text":"all 4 units to the right, something like this."},{"Start":"00:38.850 ","End":"00:41.685","Text":"Plotted the points and I joined them with a line,"},{"Start":"00:41.685 ","End":"00:44.420","Text":"and now I\u0027ve got this orange colored function."},{"Start":"00:44.420 ","End":"00:46.535","Text":"I\u0027ll just tell you what it is."},{"Start":"00:46.535 ","End":"00:49.805","Text":"The rule is that when we move 4 units to the right,"},{"Start":"00:49.805 ","End":"00:52.470","Text":"we subtract 4 from the x."},{"Start":"00:52.470 ","End":"00:58.624","Text":"This function will be y equals x minus 4 squared."},{"Start":"00:58.624 ","End":"01:00.260","Text":"You\u0027ll soon see why minus,"},{"Start":"01:00.260 ","End":"01:02.030","Text":"as many people want to put a plus here"},{"Start":"01:02.030 ","End":"01:03.230","Text":"because it goes to the right."},{"Start":"01:03.230 ","End":"01:04.100","Text":"Let\u0027s check it."},{"Start":"01:04.100 ","End":"01:06.595","Text":"For example, when x is 4,"},{"Start":"01:06.595 ","End":"01:08.240","Text":"x minus 4 is 0,"},{"Start":"01:08.240 ","End":"01:11.090","Text":"like this number was here and squared is 0."},{"Start":"01:11.090 ","End":"01:12.230","Text":"So that makes sense."},{"Start":"01:12.230 ","End":"01:15.485","Text":"Coincidentally, I noticed that these actually do cross,"},{"Start":"01:15.485 ","End":"01:21.920","Text":"might as well make use of this luck that we have the 0.24 common to both of them,"},{"Start":"01:21.920 ","End":"01:24.650","Text":"and actually write it again in black because"},{"Start":"01:24.650 ","End":"01:28.880","Text":"the same 0.24 is also on the black graph."},{"Start":"01:28.880 ","End":"01:30.440","Text":"If we put 2, 4 here,"},{"Start":"01:30.440 ","End":"01:32.285","Text":"we see 2 squared is 4."},{"Start":"01:32.285 ","End":"01:34.175","Text":"But on the orange 1,"},{"Start":"01:34.175 ","End":"01:35.460","Text":"when x is 2,"},{"Start":"01:35.460 ","End":"01:37.905","Text":"2 minus 4 is minus 2,"},{"Start":"01:37.905 ","End":"01:39.780","Text":"minus 2 squared is also 4,"},{"Start":"01:39.780 ","End":"01:41.255","Text":"so that fits also."},{"Start":"01:41.255 ","End":"01:43.160","Text":"You could even try another 1."},{"Start":"01:43.160 ","End":"01:45.275","Text":"Let\u0027s say x equals 5,"},{"Start":"01:45.275 ","End":"01:47.740","Text":"5 minus 4 is 1."},{"Start":"01:47.740 ","End":"01:50.175","Text":"1 squared is 1, works."},{"Start":"01:50.175 ","End":"01:53.750","Text":"Because you see what\u0027s happening is the minus 4 is transporting it to here."},{"Start":"01:53.750 ","End":"01:55.880","Text":"If I want to know where 5 goes,"},{"Start":"01:55.880 ","End":"01:59.940","Text":"I go 4 to the left and see where it goes on the black parabola,"},{"Start":"01:59.940 ","End":"02:01.760","Text":"and I copy that back here."},{"Start":"02:01.760 ","End":"02:04.309","Text":"In any event, you don\u0027t have to understand the reason,"},{"Start":"02:04.309 ","End":"02:06.515","Text":"just remember to do it the right way."},{"Start":"02:06.515 ","End":"02:08.480","Text":"If you put it as a plus 4,"},{"Start":"02:08.480 ","End":"02:10.580","Text":"shift 4 to the right,"},{"Start":"02:10.580 ","End":"02:13.100","Text":"instead of x, x minus 4."},{"Start":"02:13.100 ","End":"02:15.080","Text":"Now let\u0027s try going the other way,"},{"Start":"02:15.080 ","End":"02:16.425","Text":"obviously if we\u0027ve done right."},{"Start":"02:16.425 ","End":"02:18.590","Text":"Now we want to see what happens when we shift it to the left."},{"Start":"02:18.590 ","End":"02:24.580","Text":"Now where the instruction is move this parabola 5 units to the left,"},{"Start":"02:24.580 ","End":"02:28.415","Text":"so I would take these points and join them up,"},{"Start":"02:28.415 ","End":"02:30.455","Text":"say we\u0027ll do it in blue this time."},{"Start":"02:30.455 ","End":"02:32.290","Text":"I\u0027ll tell you what the equation of this is,"},{"Start":"02:32.290 ","End":"02:33.080","Text":"as you can guess,"},{"Start":"02:33.080 ","End":"02:36.170","Text":"if we moved it to the right and got a minus 4,"},{"Start":"02:36.170 ","End":"02:38.390","Text":"you\u0027re probably guessing and you\u0027d be right."},{"Start":"02:38.390 ","End":"02:42.530","Text":"This is y equals x plus 5 squared,"},{"Start":"02:42.530 ","End":"02:44.450","Text":"and we can check, for example,"},{"Start":"02:44.450 ","End":"02:47.375","Text":"if you put in x equals negative 5,"},{"Start":"02:47.375 ","End":"02:48.740","Text":"that 5 is 0,"},{"Start":"02:48.740 ","End":"02:51.335","Text":"0 squared is 0, which is just fine."},{"Start":"02:51.335 ","End":"02:55.265","Text":"Try, let\u0027s say x equals minus 4."},{"Start":"02:55.265 ","End":"02:57.935","Text":"If you add 5, it\u0027s plus 1 squared is 1,"},{"Start":"02:57.935 ","End":"02:59.330","Text":"and you\u0027ll check it all works out."},{"Start":"02:59.330 ","End":"03:04.850","Text":"The plus 5 is because I take the x value and look at the black parabola,"},{"Start":"03:04.850 ","End":"03:07.700","Text":"but I have to go 5 to the right to get to the corresponding point,"},{"Start":"03:07.700 ","End":"03:09.995","Text":"and then I look up the y and put it back here."},{"Start":"03:09.995 ","End":"03:11.450","Text":"That\u0027s the general idea."},{"Start":"03:11.450 ","End":"03:12.800","Text":"But if you don\u0027t follow that,"},{"Start":"03:12.800 ","End":"03:14.585","Text":"just remember it as a rule."},{"Start":"03:14.585 ","End":"03:17.300","Text":"First of all, we generalize from x squared to an"},{"Start":"03:17.300 ","End":"03:20.510","Text":"arbitrary function f. Let\u0027s do that here also."},{"Start":"03:20.510 ","End":"03:23.315","Text":"Supposing we just had any old function f,"},{"Start":"03:23.315 ","End":"03:26.690","Text":"if we have a function f of x and we want to move it 4 units to the right,"},{"Start":"03:26.690 ","End":"03:29.515","Text":"the new function is f of x minus 4."},{"Start":"03:29.515 ","End":"03:31.970","Text":"If we moved it 5 units to the left,"},{"Start":"03:31.970 ","End":"03:35.210","Text":"the new function is f of x plus 5."},{"Start":"03:35.210 ","End":"03:38.990","Text":"If you have a specific formula for f and we actually compute what this function is,"},{"Start":"03:38.990 ","End":"03:41.240","Text":"all this, but this is at a general level."},{"Start":"03:41.240 ","End":"03:43.070","Text":"Once again, we\u0027re also going to generalize"},{"Start":"03:43.070 ","End":"03:45.590","Text":"because someone might not have said 4 units to the right,"},{"Start":"03:45.590 ","End":"03:50.465","Text":"might\u0027ve said k units to the right and to the left might have said l units to the left."},{"Start":"03:50.465 ","End":"03:52.310","Text":"So now we can generalize,"},{"Start":"03:52.310 ","End":"03:57.065","Text":"I\u0027ve replaced the 5 with l and the k for 4."},{"Start":"03:57.065 ","End":"03:58.445","Text":"What we can say is this,"},{"Start":"03:58.445 ","End":"04:03.035","Text":"if you\u0027re given a function and you want to move it k units to the right,"},{"Start":"04:03.035 ","End":"04:08.105","Text":"you replace x by x minus k. If you want to move it, shift it,"},{"Start":"04:08.105 ","End":"04:11.450","Text":"translate it l units to the left,"},{"Start":"04:11.450 ","End":"04:14.570","Text":"then you replace x by x plus l."},{"Start":"04:14.570 ","End":"04:15.860","Text":"That\u0027s the rule."},{"Start":"04:15.860 ","End":"04:17.570","Text":"For the rest of the lesson,"},{"Start":"04:17.570 ","End":"04:20.130","Text":"we\u0027ll just do some examples."}],"ID":8403},{"Watched":false,"Name":"Translation of Functions - Examples","Duration":"5m 54s","ChapterTopicVideoID":8242,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"The first example we\u0027ll do is the following."},{"Start":"00:02.730 ","End":"00:08.025","Text":"The sketch, y equals the square root of x plus 2."},{"Start":"00:08.025 ","End":"00:11.280","Text":"How I go about this is I look at my stock of"},{"Start":"00:11.280 ","End":"00:15.240","Text":"basic functions and I see square root of x is the closest,"},{"Start":"00:15.240 ","End":"00:19.110","Text":"and I have to somehow shift it up and down to get this one."},{"Start":"00:19.110 ","End":"00:21.300","Text":"So first of all, I sketch the basic,"},{"Start":"00:21.300 ","End":"00:23.235","Text":"y equals the square root of x,"},{"Start":"00:23.235 ","End":"00:24.990","Text":"I will get something like this."},{"Start":"00:24.990 ","End":"00:27.975","Text":"Then we notice that what we have is not this,"},{"Start":"00:27.975 ","End":"00:29.650","Text":"but this plus 2."},{"Start":"00:29.650 ","End":"00:34.035","Text":"We remember that when we have a function of x plus a constant,"},{"Start":"00:34.035 ","End":"00:36.580","Text":"we just raise it upwards by 2 units,"},{"Start":"00:36.580 ","End":"00:38.475","Text":"so we\u0027ll get something like this."},{"Start":"00:38.475 ","End":"00:41.600","Text":"We take this graph of y equals the square root"},{"Start":"00:41.600 ","End":"00:45.695","Text":"of x plus 2 means raise it 2 upwards,"},{"Start":"00:45.695 ","End":"00:48.375","Text":"and we get this, and that\u0027s all there is to it."},{"Start":"00:48.375 ","End":"00:52.985","Text":"Now, onto the next exercise where it\u0027s going to be number 2."},{"Start":"00:52.985 ","End":"00:58.955","Text":"What we\u0027re going to ask for is instead of y equals the square root of x plus 2,"},{"Start":"00:58.955 ","End":"01:01.820","Text":"we want the square root of x plus 2."},{"Start":"01:01.820 ","End":"01:07.534","Text":"I remember that when we replace x by x plus a number,"},{"Start":"01:07.534 ","End":"01:10.915","Text":"it has the effect of moving it to the left."},{"Start":"01:10.915 ","End":"01:17.645","Text":"The plus 2 means that I need to take the graph part and move it 2 units to the left."},{"Start":"01:17.645 ","End":"01:19.760","Text":"I end up getting something like this,"},{"Start":"01:19.760 ","End":"01:21.350","Text":"just instead of starting at 0,"},{"Start":"01:21.350 ","End":"01:23.185","Text":"it starts at minus 2."},{"Start":"01:23.185 ","End":"01:25.350","Text":"Otherwise, looks very much the same."},{"Start":"01:25.350 ","End":"01:26.730","Text":"Next example."},{"Start":"01:26.730 ","End":"01:34.400","Text":"Sketch, y equals x minus 3 squared plus 4."},{"Start":"01:34.400 ","End":"01:36.650","Text":"Well, it looks like it\u0027s a variation on"},{"Start":"01:36.650 ","End":"01:39.995","Text":"y equals x squared but there\u0027s a lot of stuff going on here,"},{"Start":"01:39.995 ","End":"01:41.870","Text":"there\u0027s a minus 3 and a plus 4,"},{"Start":"01:41.870 ","End":"01:44.030","Text":"so let\u0027s take it one step at a time."},{"Start":"01:44.030 ","End":"01:45.335","Text":"Let\u0027s first of all,"},{"Start":"01:45.335 ","End":"01:47.440","Text":"start with y equals x squared."},{"Start":"01:47.440 ","End":"01:51.335","Text":"Here, we have y equals x squared sketched,"},{"Start":"01:51.335 ","End":"01:52.985","Text":"and we build this thing up in steps."},{"Start":"01:52.985 ","End":"02:00.065","Text":"Next step is to go for y equals x minus 3 squared and forget about the 4 for a moment."},{"Start":"02:00.065 ","End":"02:01.640","Text":"X minus 3 squared,"},{"Start":"02:01.640 ","End":"02:04.660","Text":"we replace the x by x minus 3."},{"Start":"02:04.660 ","End":"02:08.420","Text":"When we replace x by x minus a number,"},{"Start":"02:08.420 ","End":"02:12.320","Text":"it means that we move the function to the right by that number."},{"Start":"02:12.320 ","End":"02:17.015","Text":"If we take this parabola and move it 3 units to the right,"},{"Start":"02:17.015 ","End":"02:18.685","Text":"we get something like this."},{"Start":"02:18.685 ","End":"02:22.745","Text":"Finally, we\u0027ll take the plus 4 into consideration."},{"Start":"02:22.745 ","End":"02:27.260","Text":"What we want now is to go upwards by 4 units."},{"Start":"02:27.260 ","End":"02:32.460","Text":"We end up like something in the red by just taking this blue one and raising it up."},{"Start":"02:32.460 ","End":"02:35.330","Text":"The red one is the sketch of what we want."},{"Start":"02:35.330 ","End":"02:41.000","Text":"Example 4 is to sketch y equals x plus 2 cubed minus 1."},{"Start":"02:41.000 ","End":"02:43.310","Text":"If we look at our stock of basic functions,"},{"Start":"02:43.310 ","End":"02:45.590","Text":"we see that x cubed is the one that"},{"Start":"02:45.590 ","End":"02:48.350","Text":"we need because this is basically something cubed."},{"Start":"02:48.350 ","End":"02:49.745","Text":"Let\u0027s first of all,"},{"Start":"02:49.745 ","End":"02:51.470","Text":"sketch y equals x cubed."},{"Start":"02:51.470 ","End":"02:53.960","Text":"This is roughly something like this."},{"Start":"02:53.960 ","End":"02:59.030","Text":"Next step is to notice that if we just take this part here,"},{"Start":"02:59.030 ","End":"03:01.460","Text":"that\u0027s very much like y equals x cubed,"},{"Start":"03:01.460 ","End":"03:03.950","Text":"except that x is replaced by x plus 2."},{"Start":"03:03.950 ","End":"03:10.400","Text":"So what we have to do for this one is to shift in this direction by 2 units,"},{"Start":"03:10.400 ","End":"03:14.090","Text":"we get something like this where we notice that the origin has"},{"Start":"03:14.090 ","End":"03:15.950","Text":"shifted to the point minus 2,0"},{"Start":"03:15.950 ","End":"03:18.595","Text":"after we moved the whole thing 2 to the left."},{"Start":"03:18.595 ","End":"03:24.290","Text":"The next thing we have to do is to take into account this minus 1 here which"},{"Start":"03:24.290 ","End":"03:25.940","Text":"means we need to take"},{"Start":"03:25.940 ","End":"03:30.125","Text":"this blue graph and shift it one down and we get something like this."},{"Start":"03:30.125 ","End":"03:32.690","Text":"To summarize, we went from here to here by"},{"Start":"03:32.690 ","End":"03:34.610","Text":"shifting the graph 2 to the left"},{"Start":"03:34.610 ","End":"03:35.580","Text":"and we got the blue one,"},{"Start":"03:35.580 ","End":"03:37.970","Text":"and then we went from here to here by shifting"},{"Start":"03:37.970 ","End":"03:42.735","Text":"the blue one 1 unit down and got the red one, and there we are."},{"Start":"03:42.735 ","End":"03:44.320","Text":"In example number 5,"},{"Start":"03:44.320 ","End":"03:48.875","Text":"we need to sketch y equals 1 over x minus 4."},{"Start":"03:48.875 ","End":"03:51.050","Text":"If we look into our stock of basic functions,"},{"Start":"03:51.050 ","End":"03:54.035","Text":"we see that the closest thing is 1 over x."},{"Start":"03:54.035 ","End":"03:55.760","Text":"Let\u0027s sketch that first."},{"Start":"03:55.760 ","End":"03:59.055","Text":"It looks something like this but it wasn\u0027t 1 over x,"},{"Start":"03:59.055 ","End":"04:00.945","Text":"it was 1 over x minus 4."},{"Start":"04:00.945 ","End":"04:03.865","Text":"If we replace x by x minus 4,"},{"Start":"04:03.865 ","End":"04:08.420","Text":"it has the effect of moving the graph 4 units to the right,"},{"Start":"04:08.420 ","End":"04:10.880","Text":"not to the left as some people get confused."},{"Start":"04:10.880 ","End":"04:15.055","Text":"To help sketch the graph when it\u0027s moved 4 units to the right,"},{"Start":"04:15.055 ","End":"04:16.650","Text":"I make use of the axis."},{"Start":"04:16.650 ","End":"04:21.885","Text":"I notice that the graph tends to pull toward sticks to the axis,"},{"Start":"04:21.885 ","End":"04:24.440","Text":"and especially the y-axis."},{"Start":"04:24.440 ","End":"04:26.840","Text":"This part gets closer to the y-axis,"},{"Start":"04:26.840 ","End":"04:28.955","Text":"this part gets closer to the y-axis."},{"Start":"04:28.955 ","End":"04:32.090","Text":"If I take the y-axis and shift it also 4 units to"},{"Start":"04:32.090 ","End":"04:35.960","Text":"the right and get an imaginary line through x equals 4,"},{"Start":"04:35.960 ","End":"04:39.860","Text":"I can get a dotted imaginary axis,"},{"Start":"04:39.860 ","End":"04:41.660","Text":"and that helps me with the sketch."},{"Start":"04:41.660 ","End":"04:43.805","Text":"I\u0027ll get something like this."},{"Start":"04:43.805 ","End":"04:46.025","Text":"Number 6 is our last example."},{"Start":"04:46.025 ","End":"04:50.215","Text":"We have to sketch y equals 1 over x plus 4 plus 2."},{"Start":"04:50.215 ","End":"04:54.290","Text":"It\u0027s so similar to our previous example that I\u0027m not starting from scratch,"},{"Start":"04:54.290 ","End":"04:57.034","Text":"I\u0027m starting right off with 1 over x."},{"Start":"04:57.034 ","End":"04:58.820","Text":"Whereas in the previous one,"},{"Start":"04:58.820 ","End":"05:03.415","Text":"we had x minus 4 and we shifted 4 units to the right."},{"Start":"05:03.415 ","End":"05:06.255","Text":"In this one, because it\u0027s x plus 4,"},{"Start":"05:06.255 ","End":"05:11.550","Text":"the plus means that we have to take this 1 over x and shift it 4 units to the left,"},{"Start":"05:11.550 ","End":"05:13.240","Text":"we\u0027re just ignoring the 2 for the moment,"},{"Start":"05:13.240 ","End":"05:17.270","Text":"so we\u0027re just going to look at y equals 1 over x plus 4."},{"Start":"05:17.270 ","End":"05:19.100","Text":"We\u0027ll get something like this."},{"Start":"05:19.100 ","End":"05:23.015","Text":"Now, we have to take into account that we have a plus 2 here."},{"Start":"05:23.015 ","End":"05:25.940","Text":"We\u0027ll use the same method as we did before when we use"},{"Start":"05:25.940 ","End":"05:31.130","Text":"this imaginary vertical guideline by moving the y-axis 4 units to the left."},{"Start":"05:31.130 ","End":"05:33.290","Text":"This time, we\u0027ll take the x-axis"},{"Start":"05:33.290 ","End":"05:36.660","Text":"and move it 2 units up like this."},{"Start":"05:36.660 ","End":"05:43.415","Text":"Now, we could use the blue and the red as guidelines to draw in the new function,"},{"Start":"05:43.415 ","End":"05:44.720","Text":"we get something like this."},{"Start":"05:44.720 ","End":"05:47.650","Text":"Now, this red one will be what we wanted."},{"Start":"05:47.650 ","End":"05:54.600","Text":"y equals 1 over x plus 4 plus 2. We\u0027re done."}],"ID":8402},{"Watched":false,"Name":"Reflection About The X-axis","Duration":"4m 33s","ChapterTopicVideoID":8244,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this clip, I\u0027m going to talk about how to reflect a function in the x- axis."},{"Start":"00:05.100 ","End":"00:07.440","Text":"I\u0027m going to start with my usual example"},{"Start":"00:07.440 ","End":"00:09.750","Text":"of a function y equals x squared,"},{"Start":"00:09.750 ","End":"00:11.175","Text":"I\u0027ve already sketched it."},{"Start":"00:11.175 ","End":"00:14.940","Text":"What do I mean by reflecting it in the x-axis?"},{"Start":"00:14.940 ","End":"00:18.750","Text":"What I mean is imagining that the x-axis is a mirror and I want"},{"Start":"00:18.750 ","End":"00:21.030","Text":"the mirror image of this function that will"},{"Start":"00:21.030 ","End":"00:23.745","Text":"be somewhere down here in the other direction."},{"Start":"00:23.745 ","End":"00:26.340","Text":"I could just give you a very quick rough sketch,"},{"Start":"00:26.340 ","End":"00:28.250","Text":"but I\u0027d like to be a little bit more precise."},{"Start":"00:28.250 ","End":"00:31.335","Text":"I happen to draw this by taking a few sample points,"},{"Start":"00:31.335 ","End":"00:34.890","Text":"this is 1,1, this is 2,4, 0,0, and so on."},{"Start":"00:34.890 ","End":"00:36.690","Text":"What I would begin by doing would be"},{"Start":"00:36.690 ","End":"00:39.360","Text":"to draw the mirror image of these point,"},{"Start":"00:39.360 ","End":"00:40.770","Text":"so that would be here,"},{"Start":"00:40.770 ","End":"00:43.100","Text":"this would be a mirror image through here this point if I"},{"Start":"00:43.100 ","End":"00:46.560","Text":"reflected it would come out to be here."},{"Start":"00:46.560 ","End":"00:48.560","Text":"This is minus 1,1"},{"Start":"00:48.560 ","End":"00:50.225","Text":"and this would come out here,"},{"Start":"00:50.225 ","End":"00:53.015","Text":"minus 1, minus 1, minus 2,"},{"Start":"00:53.015 ","End":"00:55.880","Text":"minus 4 and then I draw a line through"},{"Start":"00:55.880 ","End":"01:00.305","Text":"this and I\u0027d get something like this green graph here."},{"Start":"01:00.305 ","End":"01:02.780","Text":"Now what\u0027s the formula of this green graph?"},{"Start":"01:02.780 ","End":"01:04.010","Text":"Well, if you notice,"},{"Start":"01:04.010 ","End":"01:05.750","Text":"anytime I take some value,"},{"Start":"01:05.750 ","End":"01:07.340","Text":"let\u0027s take this one for example,"},{"Start":"01:07.340 ","End":"01:10.220","Text":"where x was 2 and y was 4,"},{"Start":"01:10.220 ","End":"01:12.530","Text":"when I reflect the 4 in the x axis,"},{"Start":"01:12.530 ","End":"01:14.120","Text":"it becomes minus 4."},{"Start":"01:14.120 ","End":"01:15.350","Text":"In fact, any number,"},{"Start":"01:15.350 ","End":"01:16.970","Text":"any value of y I take,"},{"Start":"01:16.970 ","End":"01:18.185","Text":"if I negate it,"},{"Start":"01:18.185 ","End":"01:21.215","Text":"take it\u0027s minus or if it\u0027s minus then take the plus"},{"Start":"01:21.215 ","End":"01:24.635","Text":"and I get the mirror image in the x axis."},{"Start":"01:24.635 ","End":"01:29.180","Text":"In other words, I have to replace the y by minus y and so instead of x squared,"},{"Start":"01:29.180 ","End":"01:31.100","Text":"I get minus x squared,"},{"Start":"01:31.100 ","End":"01:34.580","Text":"this is y equals minus x squared."},{"Start":"01:34.580 ","End":"01:37.345","Text":"Again, for example, if I took x equals 2,"},{"Start":"01:37.345 ","End":"01:38.750","Text":"x squared is 4,"},{"Start":"01:38.750 ","End":"01:40.595","Text":"which gives me the point 2, 4,"},{"Start":"01:40.595 ","End":"01:42.710","Text":"but minus x squared is minus 4,"},{"Start":"01:42.710 ","End":"01:44.930","Text":"which gives me the point 2 minus 4."},{"Start":"01:44.930 ","End":"01:46.940","Text":"That\u0027s all there is to reflection,"},{"Start":"01:46.940 ","End":"01:49.010","Text":"except that it works for any function,"},{"Start":"01:49.010 ","End":"01:50.645","Text":"not just for x squared."},{"Start":"01:50.645 ","End":"01:54.170","Text":"If I replace x squared by a general f of x, like this,"},{"Start":"01:54.170 ","End":"01:55.370","Text":"y equals f of x,"},{"Start":"01:55.370 ","End":"01:59.120","Text":"then the mirror image is simply y equals minus f of x."},{"Start":"01:59.120 ","End":"02:01.010","Text":"We\u0027ll just do some examples now,"},{"Start":"02:01.010 ","End":"02:02.930","Text":"here\u0027s our first example."},{"Start":"02:02.930 ","End":"02:08.210","Text":"We have to sketch the graph of y equals minus the square root of x."},{"Start":"02:08.210 ","End":"02:09.500","Text":"According to what we learned,"},{"Start":"02:09.500 ","End":"02:11.570","Text":"if we take as a basic function,"},{"Start":"02:11.570 ","End":"02:13.505","Text":"the square root of x,"},{"Start":"02:13.505 ","End":"02:15.650","Text":"we see we just have a minus stuck in front of"},{"Start":"02:15.650 ","End":"02:18.440","Text":"it and when we have a minus stuck in front of something,"},{"Start":"02:18.440 ","End":"02:21.035","Text":"it means that we reflect it in the x-axis."},{"Start":"02:21.035 ","End":"02:23.215","Text":"First, let\u0027s draw the square root of x."},{"Start":"02:23.215 ","End":"02:25.300","Text":"This looks something as follows,"},{"Start":"02:25.300 ","End":"02:28.820","Text":"I helped myself by using some points that I plot,"},{"Start":"02:28.820 ","End":"02:32.270","Text":"like 0, 0, 1, 1, 4, 2."},{"Start":"02:32.270 ","End":"02:34.665","Text":"This one is reflected onto itself,"},{"Start":"02:34.665 ","End":"02:37.110","Text":"this one goes over here instead of being plus 1,"},{"Start":"02:37.110 ","End":"02:40.290","Text":"it goes to minus 1 and this one at 4 instead of going to"},{"Start":"02:40.290 ","End":"02:43.610","Text":"plus 2 and it goes to minus 2 and then I draw a line and I"},{"Start":"02:43.610 ","End":"02:50.780","Text":"end up with this green curve and this will be y equals minus the square root of x."},{"Start":"02:50.780 ","End":"02:54.470","Text":"Number 2 is a slightly more complicated example,"},{"Start":"02:54.470 ","End":"02:57.830","Text":"in it we have both horizontal shift,"},{"Start":"02:57.830 ","End":"03:02.325","Text":"the vertical shift, and the reflection in the x-axis all thrown together."},{"Start":"03:02.325 ","End":"03:06.950","Text":"Y equals minus x minus 4 squared plus 5."},{"Start":"03:06.950 ","End":"03:09.830","Text":"We examine our stock of basic functions and easy to"},{"Start":"03:09.830 ","End":"03:12.710","Text":"see that this is based on the x squared function."},{"Start":"03:12.710 ","End":"03:15.950","Text":"Let\u0027s start with sketch of y equals x squared."},{"Start":"03:15.950 ","End":"03:19.895","Text":"I would like to plot a few grid points first and then draw the curve through it."},{"Start":"03:19.895 ","End":"03:22.250","Text":"This is our favorite function,"},{"Start":"03:22.250 ","End":"03:25.069","Text":"y equals x squared."},{"Start":"03:25.069 ","End":"03:28.820","Text":"What we\u0027ll do now is we\u0027ll just set up x squared,"},{"Start":"03:28.820 ","End":"03:32.450","Text":"go one step further and take x minus 4 squared."},{"Start":"03:32.450 ","End":"03:38.215","Text":"We immediately recognize that the minus 4 causes it to shift 4 units to the right,"},{"Start":"03:38.215 ","End":"03:41.595","Text":"4 in this direction."},{"Start":"03:41.595 ","End":"03:44.390","Text":"What I\u0027m going to do is take these points and draw them and draw"},{"Start":"03:44.390 ","End":"03:47.975","Text":"a curve through it and I should end up with something like this."},{"Start":"03:47.975 ","End":"03:52.745","Text":"Now the minus is going to take this function that we just sketched"},{"Start":"03:52.745 ","End":"03:58.415","Text":"and reflect it about the x-axis so we\u0027ll end up getting something like this."},{"Start":"03:58.415 ","End":"04:01.130","Text":"This is what we get when we take the blue"},{"Start":"04:01.130 ","End":"04:04.685","Text":"and reflect it with the minus and we get the orange one."},{"Start":"04:04.685 ","End":"04:07.360","Text":"There\u0027s still 1 more thing that we have to do,"},{"Start":"04:07.360 ","End":"04:12.830","Text":"and that thing is to add the plus 5 that will complete the picture and as you know,"},{"Start":"04:12.830 ","End":"04:15.170","Text":"when we take a function of x and we add 5,"},{"Start":"04:15.170 ","End":"04:18.815","Text":"it has the effect of raising the whole thing 5 upwards."},{"Start":"04:18.815 ","End":"04:19.940","Text":"Just to summarize so far,"},{"Start":"04:19.940 ","End":"04:21.565","Text":"we went 4 to the right,"},{"Start":"04:21.565 ","End":"04:24.650","Text":"then we reflect it, and now we\u0027re going 5 upward,"},{"Start":"04:24.650 ","End":"04:26.315","Text":"so we\u0027ll get something like this."},{"Start":"04:26.315 ","End":"04:28.370","Text":"This is our final answer,"},{"Start":"04:28.370 ","End":"04:33.180","Text":"the red one is the one that we wanted, on to the next."}],"ID":8404},{"Watched":false,"Name":"Reflection About The Y-axis","Duration":"2m 54s","ChapterTopicVideoID":8245,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.384","Text":"In this section, we\u0027re going to study about reflection of a function in the y axis."},{"Start":"00:05.384 ","End":"00:08.460","Text":"This comes in contrast to the previous section where we"},{"Start":"00:08.460 ","End":"00:11.820","Text":"learned about reflection of a function in the x-axis."},{"Start":"00:11.820 ","End":"00:16.935","Text":"I\u0027d like to remind you of what we had when we reflected the function in the x-axis."},{"Start":"00:16.935 ","End":"00:22.050","Text":"We took each point on the graph and found its mirror image through the x-axis,"},{"Start":"00:22.050 ","End":"00:25.290","Text":"which means that it\u0027s the same distance away, but on the other side."},{"Start":"00:25.290 ","End":"00:27.105","Text":"For example, if this was the 0.4,"},{"Start":"00:27.105 ","End":"00:30.015","Text":"2, this was the 0.4, minus 2."},{"Start":"00:30.015 ","End":"00:34.795","Text":"Very similar thing happens in the case of the y-axis that\u0027s returned there."},{"Start":"00:34.795 ","End":"00:38.720","Text":"In this case, if we reflect in the y-axis in the similar way,"},{"Start":"00:38.720 ","End":"00:40.400","Text":"we\u0027ll get the following picture."},{"Start":"00:40.400 ","End":"00:45.785","Text":"The formula is y equals the square root of minus X."},{"Start":"00:45.785 ","End":"00:47.915","Text":"Let\u0027s see that it works in this case."},{"Start":"00:47.915 ","End":"00:50.600","Text":"For example, if x is minus 4,"},{"Start":"00:50.600 ","End":"00:53.210","Text":"then minus x is plus 4."},{"Start":"00:53.210 ","End":"00:54.905","Text":"The square root of 4 is 2,"},{"Start":"00:54.905 ","End":"00:57.410","Text":"this is the 0.4, 2, that\u0027s fine."},{"Start":"00:57.410 ","End":"00:59.615","Text":"If we take x to be minus 1,"},{"Start":"00:59.615 ","End":"01:01.560","Text":"minus x is plus 1,"},{"Start":"01:01.560 ","End":"01:02.760","Text":"the square root is 1."},{"Start":"01:02.760 ","End":"01:04.055","Text":"So we see it works."},{"Start":"01:04.055 ","End":"01:07.940","Text":"Let\u0027s compare this also to the reflection in the x-axis."},{"Start":"01:07.940 ","End":"01:10.190","Text":"In the case of reflection in the x-axis,"},{"Start":"01:10.190 ","End":"01:13.085","Text":"we just put minus in front of the whole function."},{"Start":"01:13.085 ","End":"01:17.090","Text":"Here we put minus only in front of the x and in general,"},{"Start":"01:17.090 ","End":"01:19.385","Text":"if the function wasn\u0027t the square root of x,"},{"Start":"01:19.385 ","End":"01:21.005","Text":"but any f of x,"},{"Start":"01:21.005 ","End":"01:23.225","Text":"what we would get would be the following."},{"Start":"01:23.225 ","End":"01:27.020","Text":"Instead of f of x, we have f of minus x,"},{"Start":"01:27.020 ","End":"01:30.905","Text":"just replace x by minus x and that\u0027s all there is to it."},{"Start":"01:30.905 ","End":"01:38.810","Text":"First example is to sketch the function y equals minus the square root of minus x plus 3."},{"Start":"01:38.810 ","End":"01:42.350","Text":"Before I get started, I\u0027d just like to make a general remark that often,"},{"Start":"01:42.350 ","End":"01:45.110","Text":"you don\u0027t actually need to do a reflection in the y-axis."},{"Start":"01:45.110 ","End":"01:48.680","Text":"If you\u0027re lucky, the function might be symmetrical about the y-axis."},{"Start":"01:48.680 ","End":"01:51.155","Text":"For example, when y equals x squared,"},{"Start":"01:51.155 ","End":"01:53.490","Text":"if you reflect it, you just get itself."},{"Start":"01:53.490 ","End":"01:56.555","Text":"That\u0027s true in general for even functions."},{"Start":"01:56.555 ","End":"02:01.055","Text":"In this example, we have to identify a basic function that we\u0027re going to start with,"},{"Start":"02:01.055 ","End":"02:04.400","Text":"and that\u0027s clearly square root of x is what we\u0027re going to choose,"},{"Start":"02:04.400 ","End":"02:06.890","Text":"only here we have the square root of minus x."},{"Start":"02:06.890 ","End":"02:09.110","Text":"Now remember that in the theory section,"},{"Start":"02:09.110 ","End":"02:11.240","Text":"just a little bit before this, we already did that,"},{"Start":"02:11.240 ","End":"02:13.640","Text":"so why don\u0027t I just copy it from there."},{"Start":"02:13.640 ","End":"02:18.050","Text":"Next thing to do is to take care of this minus that comes before."},{"Start":"02:18.050 ","End":"02:22.355","Text":"If we remember the section on the reflection in the x-axis"},{"Start":"02:22.355 ","End":"02:27.215","Text":"means that we reflect this square root of minus x in the x-axis,"},{"Start":"02:27.215 ","End":"02:28.475","Text":"and this is what we get."},{"Start":"02:28.475 ","End":"02:30.305","Text":"The graph in orange."},{"Start":"02:30.305 ","End":"02:34.910","Text":"Finally, we have to take care of the plus 3 here,"},{"Start":"02:34.910 ","End":"02:39.830","Text":"the plus 3 has the effect of raising it 3 units up,"},{"Start":"02:39.830 ","End":"02:44.020","Text":"so we need to take the orange bit and make it go up 3,"},{"Start":"02:44.020 ","End":"02:47.045","Text":"and so we\u0027ll end up with something like this."},{"Start":"02:47.045 ","End":"02:49.820","Text":"This function in red is what we\u0027re looking for."},{"Start":"02:49.820 ","End":"02:52.850","Text":"Minus the square root of minus x plus 3."},{"Start":"02:52.850 ","End":"02:55.440","Text":"We\u0027re done with this exercise."}],"ID":8405},{"Watched":false,"Name":"Exercise 1","Duration":"1m 28s","ChapterTopicVideoID":6455,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we\u0027re given a graph of a function and we"},{"Start":"00:03.060 ","End":"00:07.580","Text":"also are given 2 other graphs of the same function shifted to new positions,"},{"Start":"00:07.580 ","End":"00:10.065","Text":"and we have to find the new equations."},{"Start":"00:10.065 ","End":"00:12.570","Text":"Specifically, the original one,"},{"Start":"00:12.570 ","End":"00:14.095","Text":"that\u0027s the one in black."},{"Start":"00:14.095 ","End":"00:18.650","Text":"This one is y equals minus 2x^2."},{"Start":"00:18.650 ","End":"00:20.735","Text":"This one in green,"},{"Start":"00:20.735 ","End":"00:26.810","Text":"we can see that it\u0027s simply the original one shifted to the right by 4 units,"},{"Start":"00:26.810 ","End":"00:32.315","Text":"and the red one is shifted to the left by 7 units."},{"Start":"00:32.315 ","End":"00:33.754","Text":"Now how do we shift?"},{"Start":"00:33.754 ","End":"00:35.389","Text":"When we shift to the right,"},{"Start":"00:35.389 ","End":"00:38.395","Text":"we have to replace x with x minus 4."},{"Start":"00:38.395 ","End":"00:42.740","Text":"In other words, here we have to replace x with x minus 4."},{"Start":"00:42.740 ","End":"00:50.555","Text":"When it\u0027s left, we replace x with x plus whatever the shift is, in this case 7."},{"Start":"00:50.555 ","End":"00:52.490","Text":"We start with the original,"},{"Start":"00:52.490 ","End":"00:55.800","Text":"that\u0027s the one in black, y equals minus 2x^2."},{"Start":"00:55.930 ","End":"00:59.510","Text":"Then to compute the green one,"},{"Start":"00:59.510 ","End":"01:04.040","Text":"I\u0027ll switch to green y equals minus 2 instead of x,"},{"Start":"01:04.040 ","End":"01:06.275","Text":"x minus 4 squared."},{"Start":"01:06.275 ","End":"01:08.075","Text":"We could expand the brackets,"},{"Start":"01:08.075 ","End":"01:09.970","Text":"but this should be good enough."},{"Start":"01:09.970 ","End":"01:12.050","Text":"Next, the red one,"},{"Start":"01:12.050 ","End":"01:13.550","Text":"I\u0027ll switch to red."},{"Start":"01:13.550 ","End":"01:17.370","Text":"We get that y is equal to minus 2."},{"Start":"01:17.370 ","End":"01:20.240","Text":"This time we\u0027ll replace x with x plus 7,"},{"Start":"01:20.240 ","End":"01:22.775","Text":"x plus 7 squared."},{"Start":"01:22.775 ","End":"01:24.810","Text":"Again, I won\u0027t bother to expand the brackets,"},{"Start":"01:24.810 ","End":"01:28.500","Text":"we\u0027ll just leave the answer like this and we\u0027re done."}],"ID":6482},{"Watched":false,"Name":"Exercise 2","Duration":"1m 14s","ChapterTopicVideoID":6456,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we\u0027re given the graph of a function,"},{"Start":"00:04.050 ","End":"00:10.109","Text":"and we\u0027re also given 2 new graphs which is the original graph shifted into new positions."},{"Start":"00:10.109 ","End":"00:12.915","Text":"We have to write the equation for those new graphs."},{"Start":"00:12.915 ","End":"00:15.525","Text":"The original one is the one in black"},{"Start":"00:15.525 ","End":"00:18.420","Text":"and that\u0027s the one that\u0027s given by this equation here."},{"Start":"00:18.420 ","End":"00:23.030","Text":"This is the one which is y equals minus 2x-squared."},{"Start":"00:23.030 ","End":"00:26.400","Text":"There\u0027s a red one and the red one is gotten from"},{"Start":"00:26.400 ","End":"00:30.795","Text":"the black one by shifting it up by 2 units,"},{"Start":"00:30.795 ","End":"00:38.550","Text":"whereas the olive brown one is the same original graph shifted down 5 units."},{"Start":"00:38.550 ","End":"00:40.745","Text":"Now when we shift something up,"},{"Start":"00:40.745 ","End":"00:46.205","Text":"we simply take the equation of the function and add plus 2 to it at the end."},{"Start":"00:46.205 ","End":"00:48.155","Text":"When we move something down,"},{"Start":"00:48.155 ","End":"00:51.860","Text":"we simply subtract 5 from the original function."},{"Start":"00:51.860 ","End":"00:58.560","Text":"In other words, if the original function is y equals minus 2x-squared,"},{"Start":"00:58.560 ","End":"01:05.840","Text":"then the red one would be y equals minus 2x-squared plus 2,"},{"Start":"01:05.840 ","End":"01:12.200","Text":"and the olive brown is y equals minus 2x-squared minus 5."},{"Start":"01:12.200 ","End":"01:14.700","Text":"That\u0027s all there is to it. We\u0027re done."}],"ID":6483},{"Watched":false,"Name":"Exercise 3","Duration":"1m 11s","ChapterTopicVideoID":6457,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"In this exercise, we\u0027re given 4 equations and 4 graphs,"},{"Start":"00:05.880 ","End":"00:09.675","Text":"and we have to match which graph belongs to which equation."},{"Start":"00:09.675 ","End":"00:11.460","Text":"Start with the first one."},{"Start":"00:11.460 ","End":"00:13.860","Text":"It looks like it\u0027s based on y equals"},{"Start":"00:13.860 ","End":"00:17.790","Text":"x^2 but x is replaced by x minus 4 and has a 2 on the end."},{"Start":"00:17.790 ","End":"00:21.480","Text":"What this means is that the graph of x^2 is moved"},{"Start":"00:21.480 ","End":"00:25.395","Text":"4 units to the right and 2 units upwards."},{"Start":"00:25.395 ","End":"00:28.007","Text":"Now 4 to the right and 2 upwards,"},{"Start":"00:28.007 ","End":"00:30.030","Text":"so that\u0027s got to be the blue one."},{"Start":"00:30.030 ","End":"00:31.140","Text":"I\u0027ll just mark this,"},{"Start":"00:31.140 ","End":"00:33.645","Text":"that this belongs to number 1."},{"Start":"00:33.645 ","End":"00:35.865","Text":"In 2, similarly,"},{"Start":"00:35.865 ","End":"00:40.850","Text":"we have 1 to the right and 3 down."},{"Start":"00:40.850 ","End":"00:43.595","Text":"A quick inspection shows that it\u0027s this green one."},{"Start":"00:43.595 ","End":"00:45.790","Text":"So that would be number 2."},{"Start":"00:45.790 ","End":"00:52.739","Text":"In 3, the plus 5 means it\u0027s shifted to the left 5 units and 2 upwards."},{"Start":"00:52.739 ","End":"00:56.835","Text":"So left 5 and up 2 is this one,"},{"Start":"00:56.835 ","End":"00:58.740","Text":"that would be number 3."},{"Start":"00:58.740 ","End":"01:05.005","Text":"Finally, the plus 4 here and the minus 5 means that it was shifted to the left,"},{"Start":"01:05.005 ","End":"01:11.980","Text":"4 and down 5 and so that\u0027s got to be number 4. We\u0027re done."}],"ID":6484},{"Watched":false,"Name":"Exercise 4","Duration":"7m 21s","ChapterTopicVideoID":6458,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.355","Text":"This exercise has 8 parts."},{"Start":"00:02.355 ","End":"00:04.920","Text":"In each part, we\u0027re given an equation,"},{"Start":"00:04.920 ","End":"00:07.290","Text":"for example, y equals x cubed."},{"Start":"00:07.290 ","End":"00:11.070","Text":"Then we\u0027re given instructions on how to shift its graph,"},{"Start":"00:11.070 ","End":"00:14.380","Text":"for example, move it left 2 and down 1."},{"Start":"00:14.380 ","End":"00:18.805","Text":"What we have to do is to write the equation of the new shifted graph,"},{"Start":"00:18.805 ","End":"00:23.620","Text":"and then to sketch both the original and the shifted 1s on the axis."},{"Start":"00:23.620 ","End":"00:24.840","Text":"Let\u0027s start."},{"Start":"00:24.840 ","End":"00:26.490","Text":"We\u0027ll start with the first 1."},{"Start":"00:26.490 ","End":"00:31.445","Text":"The first 1, the original is y equals x cubed."},{"Start":"00:31.445 ","End":"00:36.470","Text":"Now, a shift of left means replace x by x plus 2,"},{"Start":"00:36.470 ","End":"00:40.150","Text":"and down simply means that we have to subtract 1 at the end."},{"Start":"00:40.150 ","End":"00:45.800","Text":"What we get is y equals x plus 2,"},{"Start":"00:45.800 ","End":"00:50.885","Text":"because left is plus and down 1 is minus 1."},{"Start":"00:50.885 ","End":"00:53.980","Text":"Now we have to just draw the graphs of each of these."},{"Start":"00:53.980 ","End":"00:55.790","Text":"Let\u0027s see, the original 1,"},{"Start":"00:55.790 ","End":"01:00.260","Text":"y equals x cubed is something like this."},{"Start":"01:00.260 ","End":"01:04.370","Text":"If we have to shift it left 2 and down 1,"},{"Start":"01:04.370 ","End":"01:06.755","Text":"it would look something like this,"},{"Start":"01:06.755 ","End":"01:08.575","Text":"and that\u0027s it for the first 1."},{"Start":"01:08.575 ","End":"01:16.115","Text":"Next, we come to part 2, y equals x^2/3."},{"Start":"01:16.115 ","End":"01:18.275","Text":"Let\u0027s first sketch the graph,"},{"Start":"01:18.275 ","End":"01:20.270","Text":"roughly something like this,"},{"Start":"01:20.270 ","End":"01:24.725","Text":"and now, let\u0027s write the equation of the shifted graph."},{"Start":"01:24.725 ","End":"01:29.000","Text":"We have to shift it right 1 and down 3."},{"Start":"01:29.000 ","End":"01:33.530","Text":"That means that y is equal."},{"Start":"01:33.530 ","End":"01:36.425","Text":"Now because of the shift right of 1,"},{"Start":"01:36.425 ","End":"01:39.160","Text":"we have to replace x by x minus 1."},{"Start":"01:39.160 ","End":"01:42.240","Text":"We get x minus 1^2/3,"},{"Start":"01:42.920 ","End":"01:48.680","Text":"and then the down 3 means that we subtract 3."},{"Start":"01:48.680 ","End":"01:51.980","Text":"The graph should look something like this,"},{"Start":"01:51.980 ","End":"01:53.510","Text":"and that\u0027s roughly it."},{"Start":"01:53.510 ","End":"01:55.715","Text":"Next, we have part 3,"},{"Start":"01:55.715 ","End":"01:59.839","Text":"y equals the square root of x."},{"Start":"01:59.839 ","End":"02:03.515","Text":"We have to move it left 1 and to move left to 1,"},{"Start":"02:03.515 ","End":"02:06.275","Text":"we simply replace x by x plus 1,"},{"Start":"02:06.275 ","End":"02:11.600","Text":"so we get y equals the square root of x plus 1"},{"Start":"02:11.600 ","End":"02:13.970","Text":"of the equation of the shifted graph."},{"Start":"02:13.970 ","End":"02:18.005","Text":"The original graph looks something like this, roughly."},{"Start":"02:18.005 ","End":"02:20.450","Text":"If we move it to the left by 1,"},{"Start":"02:20.450 ","End":"02:23.775","Text":"it will look something like this. Again, roughly."},{"Start":"02:23.775 ","End":"02:26.960","Text":"That\u0027s the graph of square root of x plus 1, and that\u0027s it."},{"Start":"02:26.960 ","End":"02:31.900","Text":"Now part 4, y equals minus the square root of x."},{"Start":"02:31.900 ","End":"02:34.345","Text":"If we shifted right 4,"},{"Start":"02:34.345 ","End":"02:38.495","Text":"it means that we replace x by x minus 4."},{"Start":"02:38.495 ","End":"02:47.560","Text":"The new graph will be y equals minus the square root of x minus 4, the new equation."},{"Start":"02:47.560 ","End":"02:49.010","Text":"Now for the graphs."},{"Start":"02:49.010 ","End":"02:52.775","Text":"Before I draw the graph of y equals minus the square root of x,"},{"Start":"02:52.775 ","End":"02:57.655","Text":"it might help if we drew the graph of y equals the square root of"},{"Start":"02:57.655 ","End":"03:03.215","Text":"x because the minus just makes it a reflection about the x-axis."},{"Start":"03:03.215 ","End":"03:04.460","Text":"1 at a time."},{"Start":"03:04.460 ","End":"03:06.905","Text":"First, y equals square root of x,"},{"Start":"03:06.905 ","End":"03:09.590","Text":"something like this, and now,"},{"Start":"03:09.590 ","End":"03:12.200","Text":"we want to reflect it about the x-axis"},{"Start":"03:12.200 ","End":"03:14.330","Text":"to get y equals minus the square root of x,"},{"Start":"03:14.330 ","End":"03:16.160","Text":"so we get something like this."},{"Start":"03:16.160 ","End":"03:21.050","Text":"In other words, we just reflected the square root of x about the x-axis and got this."},{"Start":"03:21.050 ","End":"03:25.750","Text":"Finally, we want to shift it to the right by 4 units,"},{"Start":"03:25.750 ","End":"03:27.540","Text":"we\u0027ll get something like this,"},{"Start":"03:27.540 ","End":"03:29.030","Text":"and that\u0027s the red 1."},{"Start":"03:29.030 ","End":"03:34.220","Text":"Y equals minus the square root of x minus 4."},{"Start":"03:34.220 ","End":"03:36.465","Text":"Now we come to part 5,"},{"Start":"03:36.465 ","End":"03:42.390","Text":"which is y equals 2x minus 7."},{"Start":"03:42.390 ","End":"03:43.980","Text":"If we move it up 4,"},{"Start":"03:43.980 ","End":"03:46.500","Text":"we just simply add 4 to the function,"},{"Start":"03:46.500 ","End":"03:53.765","Text":"so we get y equals 2x minus 7 plus 4,"},{"Start":"03:53.765 ","End":"03:55.115","Text":"or in other words,"},{"Start":"03:55.115 ","End":"04:00.000","Text":"y equals 2x minus 3."},{"Start":"04:00.000 ","End":"04:02.840","Text":"The first graph, it\u0027s a graph of a straight line."},{"Start":"04:02.840 ","End":"04:04.640","Text":"It\u0027s easy enough to draw."},{"Start":"04:04.640 ","End":"04:08.360","Text":"I just computed a few points and I joined them with a straight line."},{"Start":"04:08.360 ","End":"04:12.205","Text":"The next 1, which is the 1 that\u0027s shifted to the 4,"},{"Start":"04:12.205 ","End":"04:14.430","Text":"it should look something like this,"},{"Start":"04:14.430 ","End":"04:15.580","Text":"at the red 1,"},{"Start":"04:15.580 ","End":"04:17.000","Text":"which is just like the black 1,"},{"Start":"04:17.000 ","End":"04:18.790","Text":"but moved up with 4 units."},{"Start":"04:18.790 ","End":"04:20.140","Text":"In number 6,"},{"Start":"04:20.140 ","End":"04:24.230","Text":"we can use a little trick if we see the original equation,"},{"Start":"04:24.230 ","End":"04:27.160","Text":"y equals 1/2 of x plus 1 plus 2."},{"Start":"04:27.160 ","End":"04:32.090","Text":"We see it looks very much like a shift of y equals 1/2x."},{"Start":"04:32.090 ","End":"04:36.815","Text":"In fact, if we take y equals 1/2x,"},{"Start":"04:36.815 ","End":"04:45.725","Text":"we can make this into y equals 1/2 of x plus 1 plus 2,"},{"Start":"04:45.725 ","End":"04:51.360","Text":"if we go up 2 and left 1."},{"Start":"04:51.360 ","End":"04:57.210","Text":"Now, if we add to that down 5 and right 2,"},{"Start":"04:57.210 ","End":"04:59.970","Text":"we can combine these and say,"},{"Start":"04:59.970 ","End":"05:02.010","Text":"up 2, down 5,"},{"Start":"05:02.010 ","End":"05:07.605","Text":"that means down by 3 and left 1 and right 2,"},{"Start":"05:07.605 ","End":"05:10.490","Text":"is the same as right 1."},{"Start":"05:10.490 ","End":"05:14.675","Text":"If we apply this transformation to this equation,"},{"Start":"05:14.675 ","End":"05:16.190","Text":"we should get the answer."},{"Start":"05:16.190 ","End":"05:20.900","Text":"Yes, I forgot the equation of the final 1 would be to take this"},{"Start":"05:20.900 ","End":"05:25.980","Text":"and write y equals 1/2 of,"},{"Start":"05:25.980 ","End":"05:27.555","Text":"right 1 first of all,"},{"Start":"05:27.555 ","End":"05:29.970","Text":"makes it x minus 1,"},{"Start":"05:29.970 ","End":"05:34.365","Text":"and the down 3 means minus 3."},{"Start":"05:34.365 ","End":"05:38.410","Text":"This is the answer to the equation part of the end,"},{"Start":"05:38.410 ","End":"05:39.770","Text":"and now let\u0027s draw."},{"Start":"05:39.770 ","End":"05:42.650","Text":"First, we have y equals 1/2x."},{"Start":"05:42.650 ","End":"05:49.310","Text":"Next, we\u0027re going to do the original y equals 1/2 of x plus 1 plus 2."},{"Start":"05:49.310 ","End":"05:51.170","Text":"What we\u0027ll do is we\u0027ll take this,"},{"Start":"05:51.170 ","End":"05:55.335","Text":"and we\u0027ll move it up by 2 and left by 1,"},{"Start":"05:55.335 ","End":"05:58.249","Text":"so we\u0027ll get something like this in black."},{"Start":"05:58.249 ","End":"05:59.420","Text":"The last 1,"},{"Start":"05:59.420 ","End":"06:02.240","Text":"we get by taking y equals 1/2x"},{"Start":"06:02.240 ","End":"06:06.790","Text":"and making it move down 3 and right 1."},{"Start":"06:06.790 ","End":"06:08.615","Text":"We\u0027ll get something like this."},{"Start":"06:08.615 ","End":"06:11.510","Text":"That\u0027s the red graph, and we\u0027re done."},{"Start":"06:11.510 ","End":"06:16.310","Text":"Next is part 7, and I save time by pre-drawing the pictures."},{"Start":"06:16.310 ","End":"06:18.020","Text":"Hope I didn\u0027t spoil the surprise."},{"Start":"06:18.020 ","End":"06:22.350","Text":"Anyway, y equals 1/x is the black 1,"},{"Start":"06:22.350 ","End":"06:23.880","Text":"and it\u0027s a familiar graph,"},{"Start":"06:23.880 ","End":"06:26.225","Text":"that\u0027s the usual hyperbola."},{"Start":"06:26.225 ","End":"06:29.060","Text":"If we need to shift it right by 1,"},{"Start":"06:29.060 ","End":"06:31.655","Text":"we replace x with x minus 1."},{"Start":"06:31.655 ","End":"06:36.480","Text":"That part gives us 1/x minus 1,"},{"Start":"06:36.480 ","End":"06:39.445","Text":"and the up 3 means we add 3."},{"Start":"06:39.445 ","End":"06:43.820","Text":"Graphically, we just shifted to the right by 1 and up 3."},{"Start":"06:43.820 ","End":"06:47.405","Text":"For example, this point here goes to this point here and so on."},{"Start":"06:47.405 ","End":"06:49.520","Text":"Just shift, right and up."},{"Start":"06:49.520 ","End":"06:50.910","Text":"That\u0027s it for part 7."},{"Start":"06:50.910 ","End":"06:53.565","Text":"Finally, we come to part 8,"},{"Start":"06:53.565 ","End":"06:55.970","Text":"y equals 1/x squared."},{"Start":"06:55.970 ","End":"06:58.880","Text":"This has the usual shape, familiar."},{"Start":"06:58.880 ","End":"07:00.585","Text":"I\u0027ve sketched it here already."},{"Start":"07:00.585 ","End":"07:03.960","Text":"If we want to move it left by 2 and down by 3,"},{"Start":"07:03.960 ","End":"07:08.100","Text":"it becomes y equals 1 over,"},{"Start":"07:08.100 ","End":"07:09.525","Text":"now the left 2,"},{"Start":"07:09.525 ","End":"07:12.435","Text":"means we replace x by x plus 2,"},{"Start":"07:12.435 ","End":"07:15.555","Text":"and the down 3 means we subtract 3,"},{"Start":"07:15.555 ","End":"07:18.830","Text":"and if we sketch this in red,"},{"Start":"07:18.830 ","End":"07:20.360","Text":"it looks something like this."},{"Start":"07:20.360 ","End":"07:22.380","Text":"That\u0027s it."}],"ID":6485},{"Watched":false,"Name":"Exercise 5","Duration":"1m 11s","ChapterTopicVideoID":6459,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"In this exercise, we\u0027re given 2 equations and"},{"Start":"00:03.855 ","End":"00:08.850","Text":"2 graphs and we have to say which graph belongs to which equation."},{"Start":"00:08.850 ","End":"00:10.470","Text":"Now, if you look at them,"},{"Start":"00:10.470 ","End":"00:16.500","Text":"they\u0027re both based on y equals minus x squared,"},{"Start":"00:16.500 ","End":"00:18.930","Text":"and each of them is shifted left and right,"},{"Start":"00:18.930 ","End":"00:21.135","Text":"up or down in various ways."},{"Start":"00:21.135 ","End":"00:25.070","Text":"If we draw a quick sketch of y equals minus x squared,"},{"Start":"00:25.070 ","End":"00:27.660","Text":"it will look something like this."},{"Start":"00:27.660 ","End":"00:30.450","Text":"In this case, the difference is that instead of x,"},{"Start":"00:30.450 ","End":"00:32.170","Text":"we have x minus 1."},{"Start":"00:32.170 ","End":"00:38.355","Text":"This is the same graph that shifted to the right by 1 of this."},{"Start":"00:38.355 ","End":"00:41.010","Text":"The 2nd 1, we have instead of x,"},{"Start":"00:41.010 ","End":"00:45.525","Text":"x plus 2 so that means it shifted to the left 2,"},{"Start":"00:45.525 ","End":"00:47.235","Text":"and because of the plus 3,"},{"Start":"00:47.235 ","End":"00:49.530","Text":"and we have up 3."},{"Start":"00:49.530 ","End":"00:52.320","Text":"Now, if we look and see from the black 1,"},{"Start":"00:52.320 ","End":"00:56.070","Text":"which 1 is left 2 and up 3 and which 1 is right 1?"},{"Start":"00:56.070 ","End":"00:58.910","Text":"Well, clearly, this 1 has been shifted right by 1,"},{"Start":"00:58.910 ","End":"01:02.370","Text":"so that belongs to the exercise number 1."},{"Start":"01:02.370 ","End":"01:06.435","Text":"This 1 is the 1 that\u0027s shifted left 2 and up 3."},{"Start":"01:06.435 ","End":"01:12.300","Text":"This is the 1 that belongs to number 2 and that\u0027s all there is to it."}],"ID":6486},{"Watched":false,"Name":"Exercise 6","Duration":"1m 37s","ChapterTopicVideoID":6460,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.055","Text":"In this exercise, we\u0027re given 2 equations, here and here."},{"Start":"00:05.055 ","End":"00:07.755","Text":"We\u0027re given 2 graphs, here and here."},{"Start":"00:07.755 ","End":"00:10.890","Text":"We have to say which 1 belongs to which,"},{"Start":"00:10.890 ","End":"00:13.440","Text":"which equation goes with which graph."},{"Start":"00:13.440 ","End":"00:21.690","Text":"Both of these look like they\u0027re based on y equals the square root of minus x."},{"Start":"00:21.690 ","End":"00:27.055","Text":"There are similar except with some shifts to the left and right or up and down."},{"Start":"00:27.055 ","End":"00:29.990","Text":"It\u0027s easy to see this in the first"},{"Start":"00:29.990 ","End":"00:33.230","Text":"1 because it\u0027s exactly the same except for the plus 2,"},{"Start":"00:33.230 ","End":"00:39.200","Text":"which means that we take y equals square root of minus x and move it up to,"},{"Start":"00:39.200 ","End":"00:41.840","Text":"not immediately clear in the second 1."},{"Start":"00:41.840 ","End":"00:45.950","Text":"But if you write the 4 minus x in terms of something"},{"Start":"00:45.950 ","End":"00:50.465","Text":"algebraic quickly equivalent minus x minus 4."},{"Start":"00:50.465 ","End":"00:56.555","Text":"We see that this comes from this by replacing x with x minus 4."},{"Start":"00:56.555 ","End":"00:58.475","Text":"First of all, which means,"},{"Start":"00:58.475 ","End":"01:05.410","Text":"right 4 and the minus 3 means down 3."},{"Start":"01:05.410 ","End":"01:12.555","Text":"The original y equals square root of x would look something like this roughly."},{"Start":"01:12.555 ","End":"01:15.365","Text":"If I just to help make it easier,"},{"Start":"01:15.365 ","End":"01:18.020","Text":"highlight the endpoints on all 3 of them."},{"Start":"01:18.020 ","End":"01:24.035","Text":"It\u0027s quite easy to see that this 1 is the 1 which is up 2."},{"Start":"01:24.035 ","End":"01:27.415","Text":"That 1 is number 1."},{"Start":"01:27.415 ","End":"01:32.505","Text":"The 1 that\u0027s right 4 and down 3 is the lower 1."},{"Start":"01:32.505 ","End":"01:37.810","Text":"This is, goes with number 2. We\u0027re done."}],"ID":6487},{"Watched":false,"Name":"Exercise 7 - Parts 1-4","Duration":"5m 41s","ChapterTopicVideoID":6461,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"Begin with number 1,"},{"Start":"00:01.470 ","End":"00:04.710","Text":"we have y equals x squared minus 4x plus 6,"},{"Start":"00:04.710 ","End":"00:06.540","Text":"and we need to draw its graph."},{"Start":"00:06.540 ","End":"00:09.120","Text":"What I\u0027m going to do is use the method of"},{"Start":"00:09.120 ","End":"00:11.690","Text":"shifting left and right, up and down."},{"Start":"00:11.690 ","End":"00:15.240","Text":"This 1 looks like it\u0027s a variation of x squared."},{"Start":"00:15.240 ","End":"00:17.775","Text":"We\u0027ll use the method of completing the square,"},{"Start":"00:17.775 ","End":"00:22.080","Text":"so y equals x squared minus 4x."},{"Start":"00:22.080 ","End":"00:24.120","Text":"Now we have to write something here,"},{"Start":"00:24.120 ","End":"00:25.800","Text":"which makes this a perfect square."},{"Start":"00:25.800 ","End":"00:27.225","Text":"Well, this is very familiar."},{"Start":"00:27.225 ","End":"00:31.485","Text":"It\u0027s x minus 2 squared is x squared minus 4x plus 4."},{"Start":"00:31.485 ","End":"00:32.960","Text":"But there was a 6 here,"},{"Start":"00:32.960 ","End":"00:35.495","Text":"so I have to compensate by adding 2."},{"Start":"00:35.495 ","End":"00:42.895","Text":"This equals now x minus 2 squared plus 2 and the minus 2,"},{"Start":"00:42.895 ","End":"00:44.940","Text":"when you replace x with x minus 2,"},{"Start":"00:44.940 ","End":"00:46.650","Text":"this is a shift, right."},{"Start":"00:46.650 ","End":"00:50.475","Text":"I\u0027ll write, right 2 and the plus 2 at the end"},{"Start":"00:50.475 ","End":"00:55.295","Text":"means up 2 and all this is based on the x squared function."},{"Start":"00:55.295 ","End":"00:57.725","Text":"Let\u0027s start with the original we\u0027ll sketch that,"},{"Start":"00:57.725 ","End":"01:00.170","Text":"then we\u0027ll shift it right 2 and then we\u0027ll shift"},{"Start":"01:00.170 ","End":"01:02.410","Text":"it up 2 and see where we are."},{"Start":"01:02.410 ","End":"01:05.280","Text":"First of all, y equal x squared."},{"Start":"01:05.280 ","End":"01:07.070","Text":"This is the familiar function,"},{"Start":"01:07.070 ","End":"01:08.960","Text":"so I just drew a few points,"},{"Start":"01:08.960 ","End":"01:10.520","Text":"I roughly sketched the line through it."},{"Start":"01:10.520 ","End":"01:11.645","Text":"This is very familiar."},{"Start":"01:11.645 ","End":"01:17.490","Text":"The next thing to do is to shift it right by 2 units."},{"Start":"01:17.490 ","End":"01:20.225","Text":"We\u0027ll get something like this,"},{"Start":"01:20.225 ","End":"01:21.545","Text":"just a rough sketch."},{"Start":"01:21.545 ","End":"01:23.270","Text":"It should be the same shape as this."},{"Start":"01:23.270 ","End":"01:25.355","Text":"Just move 2 to the right."},{"Start":"01:25.355 ","End":"01:29.210","Text":"Next thing is to move it up to want to see"},{"Start":"01:29.210 ","End":"01:33.135","Text":"what it looks like when I move it up 2."},{"Start":"01:33.135 ","End":"01:35.530","Text":"We\u0027ll get something like this,"},{"Start":"01:35.530 ","End":"01:37.685","Text":"and we\u0027re done for part 1."},{"Start":"01:37.685 ","End":"01:40.490","Text":"Part 2, we have to sketch the graph of y"},{"Start":"01:40.490 ","End":"01:42.140","Text":"equals the square root of 4 minus"},{"Start":"01:42.140 ","End":"01:47.194","Text":"x. I\u0027ve written it below so we can scroll down a bit."},{"Start":"01:47.194 ","End":"01:50.240","Text":"Y equals the square root of 4 minus x."},{"Start":"01:50.240 ","End":"01:53.450","Text":"Now the idea is to find a more basic function"},{"Start":"01:53.450 ","End":"01:55.490","Text":"such that this is just shifting up,"},{"Start":"01:55.490 ","End":"01:56.900","Text":"down left, right of it."},{"Start":"01:56.900 ","End":"01:59.480","Text":"The function that comes to mind is"},{"Start":"01:59.480 ","End":"02:02.450","Text":"y equals the square root of minus x."},{"Start":"02:02.450 ","End":"02:03.710","Text":"This we know how to draw,"},{"Start":"02:03.710 ","End":"02:05.060","Text":"and I\u0027ll draw it in a minute."},{"Start":"02:05.060 ","End":"02:08.870","Text":"What we do here is slightly rewrite what\u0027s"},{"Start":"02:08.870 ","End":"02:11.060","Text":"under the square root sign so we can"},{"Start":"02:11.060 ","End":"02:13.265","Text":"see a minus something here."},{"Start":"02:13.265 ","End":"02:15.340","Text":"I want a minus in front."},{"Start":"02:15.340 ","End":"02:18.110","Text":"Use the usual trick of putting a minus outside"},{"Start":"02:18.110 ","End":"02:20.195","Text":"and reversing the order of the terms."},{"Start":"02:20.195 ","End":"02:23.180","Text":"It\u0027s minus x minus 4."},{"Start":"02:23.180 ","End":"02:25.280","Text":"That way if I know the graph of this,"},{"Start":"02:25.280 ","End":"02:27.080","Text":"I\u0027ll be able to get the graph of this by"},{"Start":"02:27.080 ","End":"02:29.800","Text":"shifting it 4 units to the right."},{"Start":"02:29.800 ","End":"02:32.240","Text":"This is our basic function,"},{"Start":"02:32.240 ","End":"02:37.250","Text":"and this 1 is gotten by shifting it right 4 and that\u0027s it."},{"Start":"02:37.250 ","End":"02:39.160","Text":"There is no up-down."},{"Start":"02:39.160 ","End":"02:41.960","Text":"The original 1, if you remember,"},{"Start":"02:41.960 ","End":"02:45.590","Text":"it\u0027s just y equals the square root of x because of the minus,"},{"Start":"02:45.590 ","End":"02:47.645","Text":"it\u0027s reflected to the left."},{"Start":"02:47.645 ","End":"02:50.090","Text":"In fact, it wouldn\u0027t hurt if I remind you"},{"Start":"02:50.090 ","End":"02:54.065","Text":"of y equals the square root of x."},{"Start":"02:54.065 ","End":"02:55.970","Text":"Then from here to here,"},{"Start":"02:55.970 ","End":"02:57.965","Text":"I get a reflection."},{"Start":"02:57.965 ","End":"03:01.670","Text":"We start off with the y equals the square root of x"},{"Start":"03:01.670 ","End":"03:04.220","Text":"is a rough sketch, familiar function."},{"Start":"03:04.220 ","End":"03:09.080","Text":"Next, the square root of minus x reflecting about the y axis."},{"Start":"03:09.080 ","End":"03:12.260","Text":"We\u0027ll get something like this, that\u0027s the reflection."},{"Start":"03:12.260 ","End":"03:15.295","Text":"Now we have to move it right by 4."},{"Start":"03:15.295 ","End":"03:17.760","Text":"From here to here I\u0027ll shift it 4 to the right."},{"Start":"03:17.760 ","End":"03:19.470","Text":"We\u0027ll get something like this."},{"Start":"03:19.470 ","End":"03:21.385","Text":"We\u0027re done with part 2."},{"Start":"03:21.385 ","End":"03:24.695","Text":"Now we want to sketch the graph in part 3,"},{"Start":"03:24.695 ","End":"03:27.830","Text":"y equals absolute value of x minus 4."},{"Start":"03:27.830 ","End":"03:30.545","Text":"To give ourselves some room I\u0027ve written it down below."},{"Start":"03:30.545 ","End":"03:33.305","Text":"Here it is, we look for a basic function."},{"Start":"03:33.305 ","End":"03:35.840","Text":"This is a shift of that basic function and"},{"Start":"03:35.840 ","End":"03:40.535","Text":"the obvious candidate is y equals absolute value of x."},{"Start":"03:40.535 ","End":"03:44.660","Text":"What we have here is x is replaced by x minus 4."},{"Start":"03:44.660 ","End":"03:50.090","Text":"Which means that if we take this and shift it right 4,"},{"Start":"03:50.090 ","End":"03:51.560","Text":"that will give us this 1."},{"Start":"03:51.560 ","End":"03:54.140","Text":"Now, this is a very familiar 1"},{"Start":"03:54.140 ","End":"03:57.200","Text":"and its graph looks something like this."},{"Start":"03:57.200 ","End":"04:02.780","Text":"That\u0027s this 1, that\u0027s y equals absolute value of x."},{"Start":"04:02.780 ","End":"04:06.800","Text":"If we shift it to the right 4 we simply take the point"},{"Start":"04:06.800 ","End":"04:11.420","Text":"4 here and same thing here and here,"},{"Start":"04:11.420 ","End":"04:13.160","Text":"and that\u0027s it. We\u0027re done."},{"Start":"04:13.160 ","End":"04:15.950","Text":"Now for part 4, we have to sketch the graph."},{"Start":"04:15.950 ","End":"04:19.355","Text":"This is a variation of 1 of the basic functions."},{"Start":"04:19.355 ","End":"04:21.050","Text":"We want to shift it left,"},{"Start":"04:21.050 ","End":"04:22.535","Text":"right, up, down, etc."},{"Start":"04:22.535 ","End":"04:24.890","Text":"The basic function, the most obvious"},{"Start":"04:24.890 ","End":"04:27.965","Text":"one is y equals absolute value of x."},{"Start":"04:27.965 ","End":"04:30.710","Text":"But there\u0027s a slight difficulty here apparently,"},{"Start":"04:30.710 ","End":"04:34.805","Text":"because what we would like is x minus 2 and not 2 minus x."},{"Start":"04:34.805 ","End":"04:37.505","Text":"But if you remember, in general,"},{"Start":"04:37.505 ","End":"04:41.300","Text":"we have that the absolute value of a is equal"},{"Start":"04:41.300 ","End":"04:44.335","Text":"to the absolute value of minus a."},{"Start":"04:44.335 ","End":"04:46.205","Text":"Since this is so,"},{"Start":"04:46.205 ","End":"04:50.450","Text":"we can just rewrite this as y equals"},{"Start":"04:50.450 ","End":"04:53.150","Text":"absolute value of x minus 2."},{"Start":"04:53.150 ","End":"04:54.380","Text":"I\u0027m just reversing the order,"},{"Start":"04:54.380 ","End":"04:57.560","Text":"which is putting minus, minus 3."},{"Start":"04:57.560 ","End":"05:02.240","Text":"Now if we look at this basic function and we look at this,"},{"Start":"05:02.240 ","End":"05:04.250","Text":"which is our function,"},{"Start":"05:04.250 ","End":"05:08.600","Text":"we see that x is replaced by x minus 2,"},{"Start":"05:08.600 ","End":"05:11.420","Text":"which means that the function is shifted,"},{"Start":"05:11.420 ","End":"05:15.230","Text":"right 2 and the minus 3 at the end"},{"Start":"05:15.230 ","End":"05:19.010","Text":"means that we then move it down 3."},{"Start":"05:19.010 ","End":"05:24.080","Text":"Let\u0027s start with y equals absolute value of x."},{"Start":"05:24.080 ","End":"05:27.005","Text":"Then we\u0027ll shift it right 2."},{"Start":"05:27.005 ","End":"05:30.080","Text":"I\u0027ll just mark off 2 units."},{"Start":"05:30.080 ","End":"05:33.725","Text":"Again, same thing here and here."},{"Start":"05:33.725 ","End":"05:35.810","Text":"Then we move it down 3,"},{"Start":"05:35.810 ","End":"05:38.165","Text":"so the same apex,"},{"Start":"05:38.165 ","End":"05:39.620","Text":"and that\u0027s basically it."},{"Start":"05:39.620 ","End":"05:42.000","Text":"For this part, we\u0027re done."}],"ID":6488},{"Watched":false,"Name":"Exercise 7 - Parts 5-10","Duration":"7m 35s","ChapterTopicVideoID":6462,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"Next we sketch the graph of Part 5."},{"Start":"00:03.180 ","End":"00:07.365","Text":"Y equals 1 plus the square root of x minus 3."},{"Start":"00:07.365 ","End":"00:10.845","Text":"I\u0027ve written it again below to give us some more space."},{"Start":"00:10.845 ","End":"00:15.930","Text":"This is based on y equals the square root of x,"},{"Start":"00:15.930 ","End":"00:17.250","Text":"1 of the basic functions,"},{"Start":"00:17.250 ","End":"00:20.520","Text":"and this will be a shift left or right, up or down."},{"Start":"00:20.520 ","End":"00:25.109","Text":"Notice that this will be our starting function or basic function."},{"Start":"00:25.109 ","End":"00:27.105","Text":"In order to get from here to here,"},{"Start":"00:27.105 ","End":"00:28.260","Text":"there\u0027s 2 things we do."},{"Start":"00:28.260 ","End":"00:31.530","Text":"First, we replace x by x minus 3,"},{"Start":"00:31.530 ","End":"00:36.885","Text":"and that means that we move the graph right by 3 units."},{"Start":"00:36.885 ","End":"00:39.765","Text":"The second thing is that we\u0027ve added 1 to the function."},{"Start":"00:39.765 ","End":"00:40.950","Text":"It\u0027s usually written at the end,"},{"Start":"00:40.950 ","End":"00:42.295","Text":"doesn\u0027t matter the beginning,"},{"Start":"00:42.295 ","End":"00:47.440","Text":"which means that we have to move the graph afterwards up by 1 unit."},{"Start":"00:47.440 ","End":"00:50.295","Text":"Let\u0027s do this in 2 stages."},{"Start":"00:50.295 ","End":"00:53.150","Text":"First we\u0027ll draw y equals the square root of x"},{"Start":"00:53.150 ","End":"00:56.255","Text":"should be very familiar, something like this."},{"Start":"00:56.255 ","End":"01:00.200","Text":"Next we\u0027re going to move it right 3 here,"},{"Start":"01:00.200 ","End":"01:01.730","Text":"so something like this."},{"Start":"01:01.730 ","End":"01:04.520","Text":"Then we\u0027re going to move it 1 upwards,"},{"Start":"01:04.520 ","End":"01:08.185","Text":"down here, and we end up with something like this."},{"Start":"01:08.185 ","End":"01:10.220","Text":"We\u0027re done with this part."},{"Start":"01:10.220 ","End":"01:13.280","Text":"Now we want to sketch the graph of Part 6,"},{"Start":"01:13.280 ","End":"01:15.950","Text":"y equals 3 minus square root of x."},{"Start":"01:15.950 ","End":"01:18.830","Text":"We start with a basic function and then do shifting"},{"Start":"01:18.830 ","End":"01:21.960","Text":"or reflecting to get the current function."},{"Start":"01:21.960 ","End":"01:23.490","Text":"The basic function,"},{"Start":"01:23.490 ","End":"01:28.160","Text":"the obvious choice is y equals the square root of x."},{"Start":"01:28.160 ","End":"01:32.435","Text":"Now we have to make this look a bit more like this in several stages."},{"Start":"01:32.435 ","End":"01:34.340","Text":"The first step we would do,"},{"Start":"01:34.340 ","End":"01:40.310","Text":"would be to look at y equals minus the square root of x."},{"Start":"01:40.310 ","End":"01:43.040","Text":"This from here to here,"},{"Start":"01:43.040 ","End":"01:48.325","Text":"is a reflection about the x-axis."},{"Start":"01:48.325 ","End":"01:50.749","Text":"Then from here to here,"},{"Start":"01:50.749 ","End":"01:57.245","Text":"if I slightly rewrite this as square root of x minus plus 3,"},{"Start":"01:57.245 ","End":"01:59.510","Text":"then we can see that from here to here,"},{"Start":"01:59.510 ","End":"02:05.460","Text":"we\u0027ve added 3 to the function which means that was the shift of up 3."},{"Start":"02:05.460 ","End":"02:07.610","Text":"From here to here is 2 operations,"},{"Start":"02:07.610 ","End":"02:11.705","Text":"reflection in the x-axis and a vertical shift of 3."},{"Start":"02:11.705 ","End":"02:13.100","Text":"Let\u0027s see how this works."},{"Start":"02:13.100 ","End":"02:18.470","Text":"First of all, the original y equals the square root of x, roughly like this."},{"Start":"02:18.470 ","End":"02:20.890","Text":"Next we\u0027re going to reflect in the x-axis,"},{"Start":"02:20.890 ","End":"02:22.969","Text":"so we\u0027ll get something like this."},{"Start":"02:22.969 ","End":"02:28.425","Text":"Then we want to move these 3 units upwards and we\u0027ll get something like this."},{"Start":"02:28.425 ","End":"02:30.710","Text":"That\u0027s it for this part of the question."},{"Start":"02:30.710 ","End":"02:34.220","Text":"Next, we want to graph the function in Part 7."},{"Start":"02:34.220 ","End":"02:38.960","Text":"We want to find a basic function and elementary function that this resembles."},{"Start":"02:38.960 ","End":"02:45.170","Text":"The obvious choice is y equals x to the power of 2 over 3."},{"Start":"02:45.170 ","End":"02:46.460","Text":"We see that this,"},{"Start":"02:46.460 ","End":"02:48.440","Text":"because we replaced x by x plus 4,"},{"Start":"02:48.440 ","End":"02:50.269","Text":"is simply a shift of this 1."},{"Start":"02:50.269 ","End":"02:52.880","Text":"Let\u0027s first draw this 1."},{"Start":"02:52.880 ","End":"02:54.415","Text":"Although it\u0027s elementary,"},{"Start":"02:54.415 ","End":"02:55.520","Text":"you might have forgotten."},{"Start":"02:55.520 ","End":"02:58.970","Text":"1 way to do it would best be to sketch a few points."},{"Start":"02:58.970 ","End":"03:04.625","Text":"For example, we could say that when x is 0, y is 0."},{"Start":"03:04.625 ","End":"03:06.815","Text":"When x is 1,"},{"Start":"03:06.815 ","End":"03:09.960","Text":"1 to the 2/3 is also 1."},{"Start":"03:09.960 ","End":"03:12.045","Text":"Same thing with the minus 1."},{"Start":"03:12.045 ","End":"03:15.975","Text":"Minus 1 to the 2/3 is also 1."},{"Start":"03:15.975 ","End":"03:17.930","Text":"Then if we take x equals 8,"},{"Start":"03:17.930 ","End":"03:22.440","Text":"it\u0027ll come out around number 8 to the power of 2/3 is 4,"},{"Start":"03:22.440 ","End":"03:26.970","Text":"let\u0027s say 8 somewhere over here and 4 is somewhere over here."},{"Start":"03:26.970 ","End":"03:32.785","Text":"On the other side, minus 8 and 4 roughly somewhere here."},{"Start":"03:32.785 ","End":"03:36.005","Text":"This is what the function looks like."},{"Start":"03:36.005 ","End":"03:38.600","Text":"It\u0027s symmetrical about the y-axis."},{"Start":"03:38.600 ","End":"03:40.055","Text":"there\u0027s another 1/2 here,"},{"Start":"03:40.055 ","End":"03:45.805","Text":"and that\u0027s y equals x to the 2/3 as a reminder."},{"Start":"03:45.805 ","End":"03:49.680","Text":"Now what we have is x plus 4."},{"Start":"03:49.680 ","End":"03:54.245","Text":"When we replace x with x plus 4,"},{"Start":"03:54.245 ","End":"03:59.815","Text":"it means we move the graph to the left by 4 units."},{"Start":"03:59.815 ","End":"04:03.029","Text":"If we draw that on these axes,"},{"Start":"04:03.029 ","End":"04:04.670","Text":"we\u0027ll get something like this."},{"Start":"04:04.670 ","End":"04:06.020","Text":"Could have been drawn a bit nicer,"},{"Start":"04:06.020 ","End":"04:08.480","Text":"but that\u0027s the general idea and that\u0027s the answer."},{"Start":"04:08.480 ","End":"04:12.830","Text":"Now we want to sketch the graph of the function in Part 8."},{"Start":"04:12.830 ","End":"04:18.425","Text":"Here we are, y equals x minus 1 to the 2/3 minus 2."},{"Start":"04:18.425 ","End":"04:21.670","Text":"We look for a more basic elementary function that this is"},{"Start":"04:21.670 ","End":"04:25.165","Text":"based on and as before in Part 7,"},{"Start":"04:25.165 ","End":"04:29.995","Text":"we used y equals x to the 2/3."},{"Start":"04:29.995 ","End":"04:32.345","Text":"If you refer to Part 7,"},{"Start":"04:32.345 ","End":"04:33.620","Text":"we already sketched it."},{"Start":"04:33.620 ","End":"04:39.305","Text":"It was something like this and a similar bit symmetrical on the other side,"},{"Start":"04:39.305 ","End":"04:42.980","Text":"we notice that this is a variation of this."},{"Start":"04:42.980 ","End":"04:46.805","Text":"First of all, by replacing x with x minus 1,"},{"Start":"04:46.805 ","End":"04:53.015","Text":"which means that this is a shift to the right of 1 unit."},{"Start":"04:53.015 ","End":"05:00.694","Text":"The minus 2 that we subtract at the end means that we go down 2 units."},{"Start":"05:00.694 ","End":"05:02.720","Text":"Let\u0027s do each bit separately."},{"Start":"05:02.720 ","End":"05:09.070","Text":"First of all, moving it right 1 will give us something like this here."},{"Start":"05:09.070 ","End":"05:11.350","Text":"If this was y equals x to the 2/3,"},{"Start":"05:11.350 ","End":"05:15.485","Text":"this is x minus 1 and it shifted to the right by 1."},{"Start":"05:15.485 ","End":"05:20.175","Text":"Now we need to do the downward shift by 2."},{"Start":"05:20.175 ","End":"05:23.595","Text":"Shift to the right by 1 and then the same thing"},{"Start":"05:23.595 ","End":"05:27.440","Text":"move down by 2 units and we\u0027re done for this part."},{"Start":"05:27.440 ","End":"05:30.515","Text":"Now let\u0027s sketch the graph in Part 9."},{"Start":"05:30.515 ","End":"05:32.930","Text":"From the previous exercise number 8,"},{"Start":"05:32.930 ","End":"05:35.510","Text":"we decided that it\u0027s going to be based on"},{"Start":"05:35.510 ","End":"05:39.805","Text":"y equals x to the 2/3,"},{"Start":"05:39.805 ","End":"05:42.980","Text":"which I drew then and kept the picture."},{"Start":"05:42.980 ","End":"05:45.650","Text":"This is just not written exactly in function form."},{"Start":"05:45.650 ","End":"05:52.520","Text":"I prefer to write it as y equals x to the 2/3 minus 1."},{"Start":"05:52.520 ","End":"05:54.830","Text":"Now if this is the basic function,"},{"Start":"05:54.830 ","End":"05:56.990","Text":"this is exactly the same thing except"},{"Start":"05:56.990 ","End":"05:59.030","Text":"with the minus 1 at the end."},{"Start":"05:59.030 ","End":"06:05.135","Text":"This just means that we have to shift down by 1 unit."},{"Start":"06:05.135 ","End":"06:09.420","Text":"If we take this 1 and shift it down 1,"},{"Start":"06:09.420 ","End":"06:12.825","Text":"we\u0027ll get something like this and that\u0027s it,"},{"Start":"06:12.825 ","End":"06:14.240","Text":"rough sketch, we\u0027re done."},{"Start":"06:14.240 ","End":"06:18.065","Text":"Now we want to draw the graph of the function in Exercise 10."},{"Start":"06:18.065 ","End":"06:20.870","Text":"Just like in the previous few exercises,"},{"Start":"06:20.870 ","End":"06:27.335","Text":"this is based on the primitive function y equals x to the power of 2/3."},{"Start":"06:27.335 ","End":"06:29.690","Text":"I\u0027ve done this before in Exercises 7, 8,"},{"Start":"06:29.690 ","End":"06:32.660","Text":"and 9, and this is roughly what the graph looks like."},{"Start":"06:32.660 ","End":"06:35.885","Text":"I kept it. Now how do we get from here to here?"},{"Start":"06:35.885 ","End":"06:38.750","Text":"1 way of looking at this is to rewrite it as"},{"Start":"06:38.750 ","End":"06:45.245","Text":"y equals minus x to the 2/3 plus 1."},{"Start":"06:45.245 ","End":"06:49.265","Text":"Now we can get from here to here in 2 stages."},{"Start":"06:49.265 ","End":"06:52.685","Text":"First of all, we can put a minus in front of here,"},{"Start":"06:52.685 ","End":"06:56.900","Text":"y equals minus x to the 2/3."},{"Start":"06:56.900 ","End":"06:58.715","Text":"Putting a minus in front,"},{"Start":"06:58.715 ","End":"07:02.815","Text":"this means a reflection in the x-axis."},{"Start":"07:02.815 ","End":"07:07.285","Text":"Then when we go from here to here by adding 1,"},{"Start":"07:07.285 ","End":"07:10.860","Text":"y equals minus x to the 2/3."},{"Start":"07:10.860 ","End":"07:12.800","Text":"Then we write the plus 1."},{"Start":"07:12.800 ","End":"07:17.875","Text":"This plus 1 means that we shift up by 1 unit."},{"Start":"07:17.875 ","End":"07:20.135","Text":"That\u0027s going to be in 2 steps."},{"Start":"07:20.135 ","End":"07:22.145","Text":"The starting position is here."},{"Start":"07:22.145 ","End":"07:25.040","Text":"Now the reflection in the x-axis,"},{"Start":"07:25.040 ","End":"07:26.875","Text":"something like this here."},{"Start":"07:26.875 ","End":"07:29.400","Text":"Next we have to shift it up 1."},{"Start":"07:29.400 ","End":"07:33.285","Text":"This is what we get when we shift it up 1, rough sketch."},{"Start":"07:33.285 ","End":"07:35.890","Text":"That there it is, we\u0027re done."}],"ID":6489},{"Watched":false,"Name":"Exercise 8 - Parts 1-2","Duration":"5m 9s","ChapterTopicVideoID":6463,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"This exercise has 10 parts, but in each part,"},{"Start":"00:02.970 ","End":"00:06.210","Text":"we\u0027re given a function and we have to draw its graph."},{"Start":"00:06.210 ","End":"00:07.950","Text":"Now, the 1 thing that all of these have in"},{"Start":"00:07.950 ","End":"00:11.010","Text":"common is that all these functions are based on"},{"Start":"00:11.010 ","End":"00:16.080","Text":"an elementary function which is then shifted left, right, up, down."},{"Start":"00:16.080 ","End":"00:20.070","Text":"Let\u0027s write the elementary functions alongside the original functions,"},{"Start":"00:20.070 ","End":"00:21.480","Text":"and we\u0027ll do them all at once."},{"Start":"00:21.480 ","End":"00:24.074","Text":"Then we\u0027ll draw each 1 separately."},{"Start":"00:24.074 ","End":"00:28.739","Text":"In number 1, we see clearly that it\u0027s the cube root function."},{"Start":"00:28.739 ","End":"00:37.520","Text":"So y equals cube root of x is the elementary function that we\u0027re going to base this 1 on."},{"Start":"00:37.520 ","End":"00:40.085","Text":"Here, clearly, again,"},{"Start":"00:40.085 ","End":"00:43.805","Text":"it\u0027s y equals 1 over x."},{"Start":"00:43.805 ","End":"00:49.400","Text":"Here, again, y equals 1 over x."},{"Start":"00:49.400 ","End":"00:57.830","Text":"Here, we see that it\u0027s y equals 1 over x squared and it\u0027s a shift."},{"Start":"00:57.830 ","End":"00:59.540","Text":"Whenever you see an x minus 2,"},{"Start":"00:59.540 ","End":"01:04.115","Text":"we can replace it by x and we know it\u0027s a shift in this case 2 to the right, and so on."},{"Start":"01:04.115 ","End":"01:11.150","Text":"Here also, the elementary function is y equals 1 over x squared."},{"Start":"01:11.150 ","End":"01:19.660","Text":"Here, the elementary function is y equals x to the power of 3 over 2."},{"Start":"01:19.660 ","End":"01:23.585","Text":"Here, y equals 1 over x."},{"Start":"01:23.585 ","End":"01:26.900","Text":"Here, y again is equal to 1 over x."},{"Start":"01:26.900 ","End":"01:29.479","Text":"That would be our basic elementary function."},{"Start":"01:29.479 ","End":"01:33.050","Text":"Here we have y equals 1 over x squared."},{"Start":"01:33.050 ","End":"01:38.450","Text":"Here too y equals 1 over x squared."},{"Start":"01:38.450 ","End":"01:40.595","Text":"Just a bit more shifting."},{"Start":"01:40.595 ","End":"01:42.575","Text":"That\u0027s this part."},{"Start":"01:42.575 ","End":"01:45.680","Text":"Now let\u0027s move on to the individual ones."},{"Start":"01:45.680 ","End":"01:49.204","Text":"In part 1, we have to sketch the graph of this function."},{"Start":"01:49.204 ","End":"01:52.940","Text":"Here we are below the function and the prototype."},{"Start":"01:52.940 ","End":"01:57.260","Text":"The way to get from here to here is in 2 steps."},{"Start":"01:57.260 ","End":"02:06.520","Text":"First of all, we replace x with x minus 4 and then we subtract 2."},{"Start":"02:06.520 ","End":"02:13.465","Text":"Now, what each of these means is that this tells us we have to shift right by"},{"Start":"02:13.465 ","End":"02:21.185","Text":"4 and the minus 2 at the end indicates that we go down 2 units."},{"Start":"02:21.185 ","End":"02:28.850","Text":"We\u0027ll begin with this and then we\u0027ll do a right shift here and a downward shift here."},{"Start":"02:28.850 ","End":"02:31.640","Text":"Now the cube root of x is defined for"},{"Start":"02:31.640 ","End":"02:35.690","Text":"all x and looks something like this, a rough sketch."},{"Start":"02:35.690 ","End":"02:37.430","Text":"If you forgot how to do this,"},{"Start":"02:37.430 ","End":"02:38.930","Text":"you can always plot some points."},{"Start":"02:38.930 ","End":"02:42.410","Text":"For example, you could plot the point and when x is 0,"},{"Start":"02:42.410 ","End":"02:46.025","Text":"y is 0, when x is 1, y is 1."},{"Start":"02:46.025 ","End":"02:48.815","Text":"When x is 8,"},{"Start":"02:48.815 ","End":"02:50.870","Text":"y is 2, and so on,"},{"Start":"02:50.870 ","End":"02:52.360","Text":"on the negative side."},{"Start":"02:52.360 ","End":"02:56.360","Text":"We end up with something like this for y equals cube root of x."},{"Start":"02:56.360 ","End":"02:59.090","Text":"Now, we have to shift it right, with 4 units."},{"Start":"02:59.090 ","End":"03:02.330","Text":"We end up here with something like this should be"},{"Start":"03:02.330 ","End":"03:06.055","Text":"the same shape as this just moved 4 units to the right."},{"Start":"03:06.055 ","End":"03:12.460","Text":"Now next thing we have to do is move it down to sketch this over here."},{"Start":"03:12.460 ","End":"03:13.990","Text":"Not the greatest sketch,"},{"Start":"03:13.990 ","End":"03:15.490","Text":"but you get the general idea."},{"Start":"03:15.490 ","End":"03:18.055","Text":"You take this and move it down to,"},{"Start":"03:18.055 ","End":"03:20.080","Text":"we\u0027re done for this part."},{"Start":"03:20.080 ","End":"03:23.395","Text":"Now let\u0027s draw the graph, part 2."},{"Start":"03:23.395 ","End":"03:28.600","Text":"Here they are the function and the elementary function and our task is to see how to"},{"Start":"03:28.600 ","End":"03:34.075","Text":"get from this 1 to this 1 by a series of shifts up or down, left or right."},{"Start":"03:34.075 ","End":"03:40.240","Text":"The first thing to notice is that we can replace x by x plus 1."},{"Start":"03:40.240 ","End":"03:44.470","Text":"This substitution, x goes to x plus 1,"},{"Start":"03:44.470 ","End":"03:49.750","Text":"has the graphic effect of moving it left by 1 unit."},{"Start":"03:49.750 ","End":"03:51.270","Text":"That\u0027s this part here."},{"Start":"03:51.270 ","End":"03:58.400","Text":"The plus 3 that we add onto the end has the effect of moving the graph up by 3 units."},{"Start":"03:58.400 ","End":"04:01.250","Text":"What we\u0027re going to do is sketch this 3,"},{"Start":"04:01.250 ","End":"04:03.830","Text":"then shift it left by 1 over here,"},{"Start":"04:03.830 ","End":"04:07.330","Text":"and then up by 3 over here and give us our final answer."},{"Start":"04:07.330 ","End":"04:10.940","Text":"First things first, y equals x to the 3 over 2."},{"Start":"04:10.940 ","End":"04:14.630","Text":"Here is y equals x to the 3 over 2."},{"Start":"04:14.630 ","End":"04:17.320","Text":"It\u0027s elementary, but not that elementary,"},{"Start":"04:17.320 ","End":"04:18.410","Text":"and in case you forgot,"},{"Start":"04:18.410 ","End":"04:19.970","Text":"you can always plot a few points."},{"Start":"04:19.970 ","End":"04:23.135","Text":"For example, when x is 0,"},{"Start":"04:23.135 ","End":"04:28.615","Text":"y is 0, when x is 1,1 to the 3 over 2 is 1."},{"Start":"04:28.615 ","End":"04:30.500","Text":"When x is 4,"},{"Start":"04:30.500 ","End":"04:32.375","Text":"4 to the 3 over 2,"},{"Start":"04:32.375 ","End":"04:35.260","Text":"it means the square root and then cubed, it\u0027s 8."},{"Start":"04:35.260 ","End":"04:36.995","Text":"We only need us a rough sketch."},{"Start":"04:36.995 ","End":"04:42.970","Text":"It\u0027s not defined for negative numbers because of the even number 2 in the denominator."},{"Start":"04:42.970 ","End":"04:44.885","Text":"That\u0027s the first part."},{"Start":"04:44.885 ","End":"04:49.325","Text":"Now, we have to take this and shift it left 1."},{"Start":"04:49.325 ","End":"04:51.050","Text":"We do this over here,"},{"Start":"04:51.050 ","End":"04:53.615","Text":"we get something like this, rough sketch."},{"Start":"04:53.615 ","End":"04:56.735","Text":"It\u0027s meant to look like this just shifted left 1."},{"Start":"04:56.735 ","End":"05:02.180","Text":"Now we just have to take this 1 and move it up 3 and we get something like this."},{"Start":"05:02.180 ","End":"05:05.150","Text":"All we need is a rough sketch to get the general idea."},{"Start":"05:05.150 ","End":"05:10.410","Text":"Again, move it left 1 and up 3, we\u0027re done."}],"ID":6490},{"Watched":false,"Name":"Exercise 8 - Parts 3-6","Duration":"3m 36s","ChapterTopicVideoID":6464,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"The next 4 exercises, 3, 4, 5,"},{"Start":"00:02.610 ","End":"00:06.135","Text":"and 6 are all based on the same prototype of 1 over x."},{"Start":"00:06.135 ","End":"00:08.130","Text":"We begin with number 3."},{"Start":"00:08.130 ","End":"00:11.760","Text":"This is the prototype and this is the function."},{"Start":"00:11.760 ","End":"00:15.330","Text":"The way to get from here to here is simply"},{"Start":"00:15.330 ","End":"00:19.590","Text":"to replace x with x minus 6,"},{"Start":"00:19.590 ","End":"00:20.880","Text":"that\u0027s the only difference."},{"Start":"00:20.880 ","End":"00:23.220","Text":"Whenever we do a substitution like this,"},{"Start":"00:23.220 ","End":"00:29.130","Text":"the minus 6 indicates that this is a shift right by 6 units."},{"Start":"00:29.130 ","End":"00:31.530","Text":"Let\u0027s first of all, sketch the prototype,"},{"Start":"00:31.530 ","End":"00:34.245","Text":"y equals 1 over x over here."},{"Start":"00:34.245 ","End":"00:35.745","Text":"This is roughly what it looks like,"},{"Start":"00:35.745 ","End":"00:37.170","Text":"it should be familiar to you,"},{"Start":"00:37.170 ","End":"00:39.070","Text":"it\u0027s the hyperbola,"},{"Start":"00:39.070 ","End":"00:40.490","Text":"and goes through 1,"},{"Start":"00:40.490 ","End":"00:42.470","Text":"1 and minus 1, minus 1."},{"Start":"00:42.470 ","End":"00:46.595","Text":"I\u0027ll keep this picture for the next few exercises."},{"Start":"00:46.595 ","End":"00:49.130","Text":"Now, back to the original function,"},{"Start":"00:49.130 ","End":"00:50.960","Text":"y equals 1 over x minus 6."},{"Start":"00:50.960 ","End":"00:52.910","Text":"Like I said, we just have to take this 1"},{"Start":"00:52.910 ","End":"00:55.500","Text":"and shift it 6 units to the right,"},{"Start":"00:55.500 ","End":"00:57.470","Text":"so we\u0027ll get something like this."},{"Start":"00:57.470 ","End":"00:59.090","Text":"This is meant to be the same as this,"},{"Start":"00:59.090 ","End":"01:01.870","Text":"just moved 6 units to the right. That\u0027s it."},{"Start":"01:01.870 ","End":"01:04.820","Text":"Now, let\u0027s sketch the graph of the function in part 4"},{"Start":"01:04.820 ","End":"01:07.985","Text":"which is also based on y equals 1 over x,"},{"Start":"01:07.985 ","End":"01:09.725","Text":"and here it is."},{"Start":"01:09.725 ","End":"01:13.384","Text":"I kept the graph of y equals 1 over x from before."},{"Start":"01:13.384 ","End":"01:18.560","Text":"The way to get from this to this is simply by subtracting 6,"},{"Start":"01:18.560 ","End":"01:21.230","Text":"and this is 1 of those elementary things"},{"Start":"01:21.230 ","End":"01:22.910","Text":"which just means that we take the graph"},{"Start":"01:22.910 ","End":"01:27.755","Text":"of this and move down 6 units."},{"Start":"01:27.755 ","End":"01:31.170","Text":"Start from here and then copy it here"},{"Start":"01:31.170 ","End":"01:33.195","Text":"but shift it 6 units down."},{"Start":"01:33.195 ","End":"01:34.635","Text":"We\u0027ll get something like this."},{"Start":"01:34.635 ","End":"01:35.975","Text":"Just to make things easier,"},{"Start":"01:35.975 ","End":"01:39.770","Text":"I drew the horizontal asymptote through minus 6,"},{"Start":"01:39.770 ","End":"01:41.555","Text":"just as a guideline."},{"Start":"01:41.555 ","End":"01:43.580","Text":"That\u0027s this part done."},{"Start":"01:43.580 ","End":"01:44.720","Text":"Next is number 5,"},{"Start":"01:44.720 ","End":"01:47.420","Text":"we have to draw the sketch of the graph."},{"Start":"01:47.420 ","End":"01:50.480","Text":"It\u0027s also based on y equals 1 over x, and in fact,"},{"Start":"01:50.480 ","End":"01:53.179","Text":"it\u0027s remarkably similar to the previous exercise,"},{"Start":"01:53.179 ","End":"01:56.960","Text":"only with minus x replaced by plus 5. Here it is."},{"Start":"01:56.960 ","End":"02:01.625","Text":"I kept the original prototype 1 over x together with its graph."},{"Start":"02:01.625 ","End":"02:04.640","Text":"The only difference between this and this is that here,"},{"Start":"02:04.640 ","End":"02:06.785","Text":"we\u0027ve added plus 5."},{"Start":"02:06.785 ","End":"02:10.820","Text":"Adding 5 geometrically means to take this graph"},{"Start":"02:10.820 ","End":"02:13.920","Text":"and move up by 5 units."},{"Start":"02:13.920 ","End":"02:17.000","Text":"All you have to do is make a copy of this here"},{"Start":"02:17.000 ","End":"02:19.100","Text":"and move it up 5 units."},{"Start":"02:19.100 ","End":"02:21.080","Text":"We\u0027ll get something like this."},{"Start":"02:21.080 ","End":"02:24.470","Text":"Same as this, moved up 5, and that\u0027s it."},{"Start":"02:24.470 ","End":"02:26.310","Text":"Part 6, we have to,"},{"Start":"02:26.310 ","End":"02:28.310","Text":"again, draw the graph of the function."},{"Start":"02:28.310 ","End":"02:31.795","Text":"This also is based on y equals 1 over x,"},{"Start":"02:31.795 ","End":"02:35.270","Text":"so here it is and I kept the original y equals 1"},{"Start":"02:35.270 ","End":"02:37.475","Text":"over x from the previous exercise."},{"Start":"02:37.475 ","End":"02:40.025","Text":"Now, to get from here to here,"},{"Start":"02:40.025 ","End":"02:42.020","Text":"there are 2 stages."},{"Start":"02:42.020 ","End":"02:47.595","Text":"One is replacing x by x plus 5,"},{"Start":"02:47.595 ","End":"02:50.210","Text":"and geometrically, this has the effect of"},{"Start":"02:50.210 ","End":"02:53.990","Text":"shifting the graph left by 5 units."},{"Start":"02:53.990 ","End":"02:57.350","Text":"The next thing we do was just subtract 2 from y,"},{"Start":"02:57.350 ","End":"03:01.550","Text":"and this minus 2 from here has the effect"},{"Start":"03:01.550 ","End":"03:05.105","Text":"of making the graph go down 2 units."},{"Start":"03:05.105 ","End":"03:07.565","Text":"What we have to do is start from this graph,"},{"Start":"03:07.565 ","End":"03:11.805","Text":"draw an intermediate stage which is moved left 5 units,"},{"Start":"03:11.805 ","End":"03:15.110","Text":"and then the result of that would take down 2 units."},{"Start":"03:15.110 ","End":"03:17.585","Text":"Now, moving this 1 left 5 units,"},{"Start":"03:17.585 ","End":"03:19.505","Text":"here, we get something like this."},{"Start":"03:19.505 ","End":"03:23.975","Text":"I use the minus 5 as a guideline to help me draw the sketch."},{"Start":"03:23.975 ","End":"03:28.035","Text":"Next thing to do is to take this and move it down 2."},{"Start":"03:28.035 ","End":"03:30.650","Text":"Draw that over here, it will look something like this."},{"Start":"03:30.650 ","End":"03:31.745","Text":"Just a rough sketch."},{"Start":"03:31.745 ","End":"03:34.490","Text":"Again, I used the guideline at minus 2,"},{"Start":"03:34.490 ","End":"03:36.960","Text":"and that\u0027s roughly it."}],"ID":6491},{"Watched":false,"Name":"Exercise 8 - Parts 7-10","Duration":"3m 49s","ChapterTopicVideoID":6465,"CourseChapterTopicPlaylistID":1180,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.539","Text":"In Exercise 7, we have to draw the graph of this function."},{"Start":"00:03.539 ","End":"00:08.220","Text":"We already decided it\u0027s based on 1 over x squared, that\u0027s our prototype."},{"Start":"00:08.220 ","End":"00:12.975","Text":"In fact, the following 4 exercises are all based on 1 over x squared."},{"Start":"00:12.975 ","End":"00:14.685","Text":"Let\u0027s begin with 7."},{"Start":"00:14.685 ","End":"00:17.400","Text":"Here below, I\u0027ve copied the function."},{"Start":"00:17.400 ","End":"00:20.790","Text":"This is the basic function a prototype it\u0027s based on"},{"Start":"00:20.790 ","End":"00:24.900","Text":"and I took the liberty of pre-drawing the graph of 1 over x squared,"},{"Start":"00:24.900 ","End":"00:26.324","Text":"which should be familiar."},{"Start":"00:26.324 ","End":"00:30.480","Text":"If not, you can just substitute a few points and see what it is."},{"Start":"00:30.480 ","End":"00:32.545","Text":"We have the graph of this."},{"Start":"00:32.545 ","End":"00:35.330","Text":"What\u0027s the difference between this and our function?"},{"Start":"00:35.330 ","End":"00:42.005","Text":"The difference is that we\u0027ve replaced x by x minus 2."},{"Start":"00:42.005 ","End":"00:46.100","Text":"When we do this, it has the effect graphically of moving"},{"Start":"00:46.100 ","End":"00:50.640","Text":"the graph to the right by 2 units."},{"Start":"00:50.640 ","End":"00:54.980","Text":"All we have to do is take this 1 and shift it 2 units to the right."},{"Start":"00:54.980 ","End":"00:56.675","Text":"We\u0027ll draw it over here."},{"Start":"00:56.675 ","End":"00:58.645","Text":"We\u0027ll get something like this."},{"Start":"00:58.645 ","End":"01:03.365","Text":"This is supposed to be the same as this except move 2 units to the right."},{"Start":"01:03.365 ","End":"01:06.125","Text":"That\u0027s it for Part 7."},{"Start":"01:06.125 ","End":"01:07.915","Text":"Now Part 8,"},{"Start":"01:07.915 ","End":"01:10.114","Text":"we have to draw the graph of this function."},{"Start":"01:10.114 ","End":"01:14.810","Text":"Once again, it\u0027s based on y equals 1 over x squared. Here it is."},{"Start":"01:14.810 ","End":"01:18.440","Text":"Here\u0027s the basic function prototype it was based on."},{"Start":"01:18.440 ","End":"01:21.935","Text":"The difference is that in our function,"},{"Start":"01:21.935 ","End":"01:23.645","Text":"we have a minus 2,"},{"Start":"01:23.645 ","End":"01:29.750","Text":"which has the effect of lowering the whole function or its graph by 2 units."},{"Start":"01:29.750 ","End":"01:35.055","Text":"Basically we start from here and we just go down 2."},{"Start":"01:35.055 ","End":"01:39.110","Text":"Other words, I have to take this picture and pull it down 2 units,"},{"Start":"01:39.110 ","End":"01:40.895","Text":"and I\u0027ll draw it over here."},{"Start":"01:40.895 ","End":"01:42.770","Text":"I\u0027ll get something like this,"},{"Start":"01:42.770 ","End":"01:44.705","Text":"rough sketch and we\u0027re done."},{"Start":"01:44.705 ","End":"01:46.550","Text":"In Number 9, as usual,"},{"Start":"01:46.550 ","End":"01:48.725","Text":"we have to draw the graph of the function."},{"Start":"01:48.725 ","End":"01:54.655","Text":"It\u0027s also based on y equals 1 over x squared and is remarkably similar to Number 8."},{"Start":"01:54.655 ","End":"01:58.820","Text":"Here it is and here\u0027s the original y equals 1 over x squared,"},{"Start":"01:58.820 ","End":"02:00.965","Text":"which it\u0027s based on the prototype."},{"Start":"02:00.965 ","End":"02:03.650","Text":"I\u0027ve kept the sketch, that 1 too."},{"Start":"02:03.650 ","End":"02:06.940","Text":"As before, the only difference between this 1 and"},{"Start":"02:06.940 ","End":"02:11.000","Text":"this 1 is that we take this and we add a 4 to it."},{"Start":"02:11.000 ","End":"02:18.825","Text":"Adding 4 to a function has the effect graphically of moving it up 4 units."},{"Start":"02:18.825 ","End":"02:22.235","Text":"What we have to do is take this and move it up 4 units,"},{"Start":"02:22.235 ","End":"02:23.885","Text":"and we\u0027ll draw that over here."},{"Start":"02:23.885 ","End":"02:25.895","Text":"It looks something like this."},{"Start":"02:25.895 ","End":"02:28.865","Text":"We just needed a rough sketch and we\u0027re done."},{"Start":"02:28.865 ","End":"02:30.950","Text":"Finally we come to Number 10."},{"Start":"02:30.950 ","End":"02:34.009","Text":"We have to sketch the graph of this function."},{"Start":"02:34.009 ","End":"02:38.660","Text":"Once again, it\u0027s based on the y equals 1 over x squared prototype."},{"Start":"02:38.660 ","End":"02:40.970","Text":"Here it is, the prototype,"},{"Start":"02:40.970 ","End":"02:44.135","Text":"1 over x squared which has the sketch here."},{"Start":"02:44.135 ","End":"02:49.505","Text":"Now, how do we get from 1 over x squared to 1 over x plus 4 squared plus 2?"},{"Start":"02:49.505 ","End":"02:51.920","Text":"Well, there\u0027s 2 steps."},{"Start":"02:51.920 ","End":"02:59.490","Text":"1 is changing x into x plus 4, I mean substituting."},{"Start":"02:59.490 ","End":"03:05.495","Text":"This has the effect of moving the graph left by 4 units."},{"Start":"03:05.495 ","End":"03:08.930","Text":"The other bit is the plus 2 on the end,"},{"Start":"03:08.930 ","End":"03:15.615","Text":"this plus 2 has the effect of making the graph go up 2 units."},{"Start":"03:15.615 ","End":"03:19.230","Text":"What we\u0027re going to do is start from this 1 over x squared,"},{"Start":"03:19.230 ","End":"03:22.725","Text":"we\u0027re going to move it left on these axis,"},{"Start":"03:22.725 ","End":"03:26.855","Text":"and then down here we\u0027ll take the result of this and move it up 2."},{"Start":"03:26.855 ","End":"03:31.020","Text":"Here it goes, first of all left 4, something like this."},{"Start":"03:31.020 ","End":"03:35.405","Text":"I used minus 4 vertical line as the asymptote to help the sketch."},{"Start":"03:35.405 ","End":"03:38.900","Text":"Next thing is to take this and move it up 2,"},{"Start":"03:38.900 ","End":"03:41.585","Text":"drawing it over here we get roughly something like this."},{"Start":"03:41.585 ","End":"03:47.870","Text":"Once again, I use the horizontal 2 asymptote as a guideline to help me draw the graph."},{"Start":"03:47.870 ","End":"03:50.490","Text":"That\u0027s it, we\u0027re done."}],"ID":6492}],"Thumbnail":null,"ID":1180},{"Name":"The Composition of Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Function Composition","Duration":"11m 15s","ChapterTopicVideoID":1244,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this clip, I hope to explain the composition of functions."},{"Start":"00:04.170 ","End":"00:06.420","Text":"Let\u0027s start with an example."},{"Start":"00:06.420 ","End":"00:13.500","Text":"f of x is equal to 4x squared plus 10x plus 1."},{"Start":"00:13.500 ","End":"00:17.955","Text":"Now, x is a placeholder and I can substitute things into it."},{"Start":"00:17.955 ","End":"00:23.235","Text":"For example, I could substitute a number like 0 and get"},{"Start":"00:23.235 ","End":"00:29.475","Text":"that f of 0 is 4 times 0 squared plus 10 times 0 plus 1."},{"Start":"00:29.475 ","End":"00:30.720","Text":"If I compute this,"},{"Start":"00:30.720 ","End":"00:32.205","Text":"it comes out 1."},{"Start":"00:32.205 ","End":"00:34.140","Text":"I could substitute a different value,"},{"Start":"00:34.140 ","End":"00:35.880","Text":"let\u0027s say f of 1,"},{"Start":"00:35.880 ","End":"00:41.960","Text":"and I\u0027d get 4 times 1 squared plus 10 times 1 plus 1,"},{"Start":"00:41.960 ","End":"00:44.765","Text":"and this time it would come out to be 15."},{"Start":"00:44.765 ","End":"00:46.910","Text":"I could even substitute letters."},{"Start":"00:46.910 ","End":"00:55.800","Text":"For example, I could say that f of a is 4a squared plus 10a plus 1."},{"Start":"00:55.800 ","End":"00:57.700","Text":"But that\u0027s not all I can substitute,"},{"Start":"00:57.700 ","End":"01:00.575","Text":"I can substitute even another function."},{"Start":"01:00.575 ","End":"01:04.250","Text":"For example, I could substitute the function 1 over x,"},{"Start":"01:04.250 ","End":"01:11.745","Text":"say f of 1 over x is equal to 4 times 1 over x squared."},{"Start":"01:11.745 ","End":"01:13.460","Text":"You see, everywhere there was x,"},{"Start":"01:13.460 ","End":"01:15.385","Text":"I\u0027m now putting 1 over x."},{"Start":"01:15.385 ","End":"01:22.715","Text":"4 times 1 over x squared plus 10 times 1 over x plus 1."},{"Start":"01:22.715 ","End":"01:26.180","Text":"If I do a bit of highlighting, wherever I had x,"},{"Start":"01:26.180 ","End":"01:28.100","Text":"I substitute it in this case,"},{"Start":"01:28.100 ","End":"01:29.465","Text":"1 over x,"},{"Start":"01:29.465 ","End":"01:31.265","Text":"and the same thing in the previous cases."},{"Start":"01:31.265 ","End":"01:34.040","Text":"Here I substituted 0 everywhere where there was x,"},{"Start":"01:34.040 ","End":"01:35.870","Text":"here I substituted 1,"},{"Start":"01:35.870 ","End":"01:37.975","Text":"and here I substituted a."},{"Start":"01:37.975 ","End":"01:40.700","Text":"Here I haven\u0027t finished the exercise yet because I can"},{"Start":"01:40.700 ","End":"01:43.190","Text":"actually expand this and say that this is"},{"Start":"01:43.190 ","End":"01:52.230","Text":"equal to 4 over x squared plus 10 over x plus 1."},{"Start":"01:52.230 ","End":"01:54.770","Text":"What I\u0027ve got is a new function."},{"Start":"01:54.770 ","End":"01:58.445","Text":"This is what we mean by the composition of functions."},{"Start":"01:58.445 ","End":"02:02.045","Text":"Now, let me take it a bit further and introduce some notation."},{"Start":"02:02.045 ","End":"02:10.055","Text":"1 over x is a function and I could have given it a name and said that g of x is 1 over x."},{"Start":"02:10.055 ","End":"02:13.235","Text":"In which case, what I have written here,"},{"Start":"02:13.235 ","End":"02:15.290","Text":"this f of 1 over x,"},{"Start":"02:15.290 ","End":"02:21.825","Text":"is actually f of g of x and that\u0027s what this equals."},{"Start":"02:21.825 ","End":"02:28.025","Text":"That\u0027s what composition is and this is called f composed with g of x."},{"Start":"02:28.025 ","End":"02:30.005","Text":"Let\u0027s do another example."},{"Start":"02:30.005 ","End":"02:32.150","Text":"This time instead of 1 over x,"},{"Start":"02:32.150 ","End":"02:33.890","Text":"let\u0027s take another function."},{"Start":"02:33.890 ","End":"02:38.700","Text":"Let\u0027s take say, h of x to be x"},{"Start":"02:38.700 ","End":"02:45.755","Text":"squared and let\u0027s see if we can compute what is f of h of x."},{"Start":"02:45.755 ","End":"02:48.380","Text":"Well, this is equal to f of,"},{"Start":"02:48.380 ","End":"02:50.785","Text":"h of x is x squared."},{"Start":"02:50.785 ","End":"02:54.450","Text":"As before, we put x squared instead of x,"},{"Start":"02:54.450 ","End":"02:55.950","Text":"x is really just a placeholder,"},{"Start":"02:55.950 ","End":"02:58.630","Text":"we put x squared instead everywhere."},{"Start":"02:58.630 ","End":"03:01.430","Text":"I\u0027m looking at the top line as I\u0027m writing."},{"Start":"03:01.430 ","End":"03:10.785","Text":"We get 4 times x squared squared plus 10 times x squared plus 1,"},{"Start":"03:10.785 ","End":"03:15.660","Text":"which is equal to 4x^4,"},{"Start":"03:15.660 ","End":"03:23.915","Text":"plus 10x squared plus 1 and this is what f composed with h of x gives me."},{"Start":"03:23.915 ","End":"03:26.660","Text":"I\u0027d like to write down the term just so you\u0027ll have it"},{"Start":"03:26.660 ","End":"03:29.705","Text":"in front of you that this is called,"},{"Start":"03:29.705 ","End":"03:31.475","Text":"in this particular case,"},{"Start":"03:31.475 ","End":"03:37.160","Text":"the composition of f with g. This of course,"},{"Start":"03:37.160 ","End":"03:40.280","Text":"will be the composition of f with h. Now,"},{"Start":"03:40.280 ","End":"03:43.085","Text":"there\u0027s another notation I want to introduce."},{"Start":"03:43.085 ","End":"03:44.960","Text":"If I look at this,"},{"Start":"03:44.960 ","End":"03:48.940","Text":"I have f of x equals some expression involving x,"},{"Start":"03:48.940 ","End":"03:51.740","Text":"but I can also talk about just f on its own"},{"Start":"03:51.740 ","End":"03:54.845","Text":"without discussing the variable or the placeholder,"},{"Start":"03:54.845 ","End":"03:57.125","Text":"and I want to do a similar thing here."},{"Start":"03:57.125 ","End":"04:03.460","Text":"On the 1 hand, it\u0027s f of g of x and it has an expression involving x,"},{"Start":"04:03.460 ","End":"04:06.980","Text":"but if I want to just refer to the function without the x,"},{"Start":"04:06.980 ","End":"04:11.990","Text":"what we do is instead of f of g of x,"},{"Start":"04:11.990 ","End":"04:16.400","Text":"we write as if f and the g were stuck together,"},{"Start":"04:16.400 ","End":"04:20.285","Text":"we write it as f composed with g,"},{"Start":"04:20.285 ","End":"04:24.605","Text":"that\u0027s f with a little circle and the g of x."},{"Start":"04:24.605 ","End":"04:27.860","Text":"This thing is like 1 function which was built"},{"Start":"04:27.860 ","End":"04:32.930","Text":"from f and g by first applying g and then f. If we want,"},{"Start":"04:32.930 ","End":"04:38.480","Text":"we can also write down what the function is in the sense that it\u0027s equal"},{"Start":"04:38.480 ","End":"04:44.525","Text":"to 4 over x squared plus 10 over x plus 1."},{"Start":"04:44.525 ","End":"04:47.060","Text":"But the fact that I\u0027ve written it this way means that instead of x,"},{"Start":"04:47.060 ","End":"04:50.075","Text":"I could put something else like a number or another latter."},{"Start":"04:50.075 ","End":"04:54.775","Text":"For instance, let\u0027s let x equals 2 in here."},{"Start":"04:54.775 ","End":"04:59.375","Text":"Now, I want to emphasize that this is now a new function,"},{"Start":"04:59.375 ","End":"05:01.910","Text":"f composed with g is a single function,"},{"Start":"05:01.910 ","End":"05:03.635","Text":"and we\u0027re going to let x equal 2."},{"Start":"05:03.635 ","End":"05:08.105","Text":"We\u0027re going to compute what is f compose g,"},{"Start":"05:08.105 ","End":"05:10.930","Text":"which is a new function of 2?"},{"Start":"05:10.930 ","End":"05:13.039","Text":"Let\u0027s see what we get."},{"Start":"05:13.039 ","End":"05:15.530","Text":"We just substitute 2 in here."},{"Start":"05:15.530 ","End":"05:25.350","Text":"4 over 2 squared plus 10 over 2 plus 1."},{"Start":"05:25.350 ","End":"05:28.380","Text":"2 squared is 4, 4 over 4 is 1,"},{"Start":"05:28.380 ","End":"05:29.790","Text":"10 over 2 is 5,"},{"Start":"05:29.790 ","End":"05:32.750","Text":"1 plus 5 plus 1 is 7."},{"Start":"05:32.750 ","End":"05:35.925","Text":"But I want to emphasize the tie-in,"},{"Start":"05:35.925 ","End":"05:42.140","Text":"this close connection between f compose g of x and f of g of x."},{"Start":"05:42.140 ","End":"05:47.120","Text":"Let\u0027s try using this expression and substituting the 2 here."},{"Start":"05:47.120 ","End":"05:53.120","Text":"To contrast, if I say f of g of 2,"},{"Start":"05:53.120 ","End":"05:56.055","Text":"this is equal to f of,"},{"Start":"05:56.055 ","End":"05:57.510","Text":"now, what is g of 2?"},{"Start":"05:57.510 ","End":"06:00.440","Text":"I\u0027ll go look for the definition of g and here it is."},{"Start":"06:00.440 ","End":"06:02.330","Text":"g of x is 1 over x,"},{"Start":"06:02.330 ","End":"06:05.400","Text":"so g of 2 is 1 over 2."},{"Start":"06:05.930 ","End":"06:09.260","Text":"Now, I\u0027ve forgotten what f of x is."},{"Start":"06:09.260 ","End":"06:17.415","Text":"f of x equals 4x squared plus 10x plus 1."},{"Start":"06:17.415 ","End":"06:22.715","Text":"Just putting x equals 1/2 in this original formula for f,"},{"Start":"06:22.715 ","End":"06:32.800","Text":"we get 4 times 1/2 squared plus 10 times 1/2 plus 1."},{"Start":"06:32.800 ","End":"06:34.740","Text":"Again, I get the same thing."},{"Start":"06:34.740 ","End":"06:35.940","Text":"A 1/2 squared is a 1/4,"},{"Start":"06:35.940 ","End":"06:37.635","Text":"times 4 is 1,"},{"Start":"06:37.635 ","End":"06:39.330","Text":"10 times a 1/2 is 5,"},{"Start":"06:39.330 ","End":"06:41.940","Text":"and this is 1 and I also get 7."},{"Start":"06:41.940 ","End":"06:43.935","Text":"I had to get the same answer,"},{"Start":"06:43.935 ","End":"06:46.335","Text":"but this is like applying 2 functions."},{"Start":"06:46.335 ","End":"06:49.515","Text":"I took 2, applied g and got 1/2,"},{"Start":"06:49.515 ","End":"06:51.885","Text":"then applied f on that and got 7."},{"Start":"06:51.885 ","End":"06:53.670","Text":"This was like an all in 1."},{"Start":"06:53.670 ","End":"06:57.690","Text":"This formula is the all in 1 for doing first g and then"},{"Start":"06:57.690 ","End":"07:02.240","Text":"f. I\u0027d like to do some more examples of composition of 2 functions,"},{"Start":"07:02.240 ","End":"07:04.625","Text":"so we\u0027ll move to the next page."},{"Start":"07:04.625 ","End":"07:07.820","Text":"I\u0027ve already written 2 functions, f,"},{"Start":"07:07.820 ","End":"07:11.735","Text":"where f of x is x plus 1 over 4x minus 8,"},{"Start":"07:11.735 ","End":"07:15.645","Text":"and g, where g of x is x squared plus 5."},{"Start":"07:15.645 ","End":"07:18.440","Text":"Actually, I\u0027ll give you 2 exercises at once."},{"Start":"07:18.440 ","End":"07:26.435","Text":"The first exercise is to compute what is f composed g. The second exercise,"},{"Start":"07:26.435 ","End":"07:30.710","Text":"compute what is g composed f. Well,"},{"Start":"07:30.710 ","End":"07:32.610","Text":"we don\u0027t have to compute a function in the abstract,"},{"Start":"07:32.610 ","End":"07:34.040","Text":"we take a dummy variable,"},{"Start":"07:34.040 ","End":"07:36.110","Text":"a place holder variable such as x."},{"Start":"07:36.110 ","End":"07:39.365","Text":"Let\u0027s see what this equals and let\u0027s see what this equals."},{"Start":"07:39.365 ","End":"07:41.720","Text":"What do you think, will we get the same thing or not?"},{"Start":"07:41.720 ","End":"07:43.130","Text":"Start with this 1."},{"Start":"07:43.130 ","End":"07:44.690","Text":"According to the definition,"},{"Start":"07:44.690 ","End":"07:50.870","Text":"f composed g of x is really f of g of x."},{"Start":"07:50.870 ","End":"07:56.600","Text":"What this equals is we take f and instead of g of x,"},{"Start":"07:56.600 ","End":"07:58.490","Text":"now I\u0027ll put what it equals,"},{"Start":"07:58.490 ","End":"08:01.250","Text":"g of x is x squared plus 5."},{"Start":"08:01.250 ","End":"08:04.580","Text":"What I have to do is everywhere in f,"},{"Start":"08:04.580 ","End":"08:07.820","Text":"where I see x in f of x,"},{"Start":"08:07.820 ","End":"08:12.650","Text":"I\u0027m going to replace it by x squared plus 5."},{"Start":"08:12.650 ","End":"08:15.660","Text":"I have x here and here,"},{"Start":"08:15.660 ","End":"08:19.175","Text":"so both of these have to be replaced by x squared plus 5."},{"Start":"08:19.175 ","End":"08:20.630","Text":"Let\u0027s get to it."},{"Start":"08:20.630 ","End":"08:27.075","Text":"f of x squared plus 5 is x squared plus 5 plus"},{"Start":"08:27.075 ","End":"08:33.585","Text":"1 over 4 times x squared plus 5 minus 8."},{"Start":"08:33.585 ","End":"08:39.900","Text":"This equals, the numerator is x squared plus 6 and on the denominator"},{"Start":"08:39.900 ","End":"08:47.940","Text":"4x squared plus 4 times 5 is 20 but less 8 means it\u0027s only plus 12."},{"Start":"08:47.940 ","End":"08:52.095","Text":"This is what f compose g does to x."},{"Start":"08:52.095 ","End":"08:54.010","Text":"Let\u0027s do the other 1."},{"Start":"08:54.010 ","End":"08:56.195","Text":"G composed f,"},{"Start":"08:56.195 ","End":"08:59.300","Text":"I should really have written this in the brackets to emphasize this is"},{"Start":"08:59.300 ","End":"09:03.085","Text":"1 function which is made up from 2 different functions."},{"Start":"09:03.085 ","End":"09:09.090","Text":"What this equals is g of f of x and this"},{"Start":"09:09.090 ","End":"09:16.840","Text":"equals g of f of x is x plus 1 over 4x minus 8."},{"Start":"09:16.840 ","End":"09:22.055","Text":"This time, we need to substitute this into g. What we get"},{"Start":"09:22.055 ","End":"09:27.094","Text":"is the turquoise thing squared plus 5 and this equals,"},{"Start":"09:27.094 ","End":"09:30.330","Text":"I\u0027m not going to bother expanding all this."},{"Start":"09:30.330 ","End":"09:33.455","Text":"It certainly looks like we got 2 different things."},{"Start":"09:33.455 ","End":"09:37.460","Text":"On the 1 hand, we got for f composed with g of x,"},{"Start":"09:37.460 ","End":"09:39.890","Text":"we got this expression."},{"Start":"09:39.890 ","End":"09:43.355","Text":"On the other hand, for g composed f of x,"},{"Start":"09:43.355 ","End":"09:46.370","Text":"we got a different expression which goes to show you that it"},{"Start":"09:46.370 ","End":"09:50.060","Text":"makes a difference in which order you do the composition of functions."},{"Start":"09:50.060 ","End":"09:56.585","Text":"G composed with f is not the same as f composed with g. Our next exercise is this."},{"Start":"09:56.585 ","End":"09:58.475","Text":"We\u0027re given f of x,"},{"Start":"09:58.475 ","End":"10:01.205","Text":"which is equal to x squared plus x plus 1,"},{"Start":"10:01.205 ","End":"10:02.810","Text":"and we\u0027re given h of x,"},{"Start":"10:02.810 ","End":"10:07.460","Text":"which is natural log of x over e^x plus 1,"},{"Start":"10:07.460 ","End":"10:11.470","Text":"and we have to compute what h composed with f is."},{"Start":"10:11.470 ","End":"10:14.825","Text":"In other words, what is h composed with f of x?"},{"Start":"10:14.825 ","End":"10:16.625","Text":"According to the definition,"},{"Start":"10:16.625 ","End":"10:22.205","Text":"this is equal to h of f of x."},{"Start":"10:22.205 ","End":"10:28.085","Text":"This equals, we take h and what\u0027s inside the bracket is replaced with f of x,"},{"Start":"10:28.085 ","End":"10:29.555","Text":"and f of x is here."},{"Start":"10:29.555 ","End":"10:31.690","Text":"It\u0027s x squared plus x plus 1."},{"Start":"10:31.690 ","End":"10:34.640","Text":"Now, how do I substitute something into h?"},{"Start":"10:34.640 ","End":"10:36.845","Text":"Wherever I saw x before,"},{"Start":"10:36.845 ","End":"10:39.425","Text":"I now put x squared plus x plus 1."},{"Start":"10:39.425 ","End":"10:42.320","Text":"What I have to do is in the definition of h,"},{"Start":"10:42.320 ","End":"10:45.290","Text":"replace each x and there\u0027s 1 here and 1 here,"},{"Start":"10:45.290 ","End":"10:48.770","Text":"and replace them with x squared plus x plus 1."},{"Start":"10:48.770 ","End":"10:57.060","Text":"What I get is the natural log of x squared plus x plus 1,"},{"Start":"10:57.060 ","End":"11:01.800","Text":"and all this over e to the power of, instead of this x,"},{"Start":"11:01.800 ","End":"11:06.090","Text":"I\u0027ll get x squared plus x plus 1,"},{"Start":"11:06.090 ","End":"11:08.185","Text":"but still plus 1."},{"Start":"11:08.185 ","End":"11:11.510","Text":"That\u0027s the answer to h composed with f. This"},{"Start":"11:11.510 ","End":"11:16.410","Text":"concludes the lesson on the composition of functions."}],"ID":1244},{"Watched":false,"Name":"Exercise 1","Duration":"1m 7s","ChapterTopicVideoID":4347,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.450 ","End":"00:06.660","Text":"odd or neither and to give reasons for our answer."},{"Start":"00:06.660 ","End":"00:08.340","Text":"Here\u0027s the function."},{"Start":"00:08.340 ","End":"00:17.940","Text":"Recall that an even function is one for which f of minus x is equal to f of x for all x,"},{"Start":"00:17.940 ","End":"00:20.455","Text":"so this is what we call even."},{"Start":"00:20.455 ","End":"00:28.650","Text":"An odd function is one for which f of minus x is equal to minus f of x for all x."},{"Start":"00:28.650 ","End":"00:30.330","Text":"That\u0027s odd."},{"Start":"00:30.330 ","End":"00:32.670","Text":"In our case, on every case,"},{"Start":"00:32.670 ","End":"00:35.985","Text":"we have to substitute minus x instead of x ."},{"Start":"00:35.985 ","End":"00:39.764","Text":"We get f of minus x,"},{"Start":"00:39.764 ","End":"00:42.295","Text":"which is to substitute minus x here,"},{"Start":"00:42.295 ","End":"00:48.420","Text":"equals 4 minus x cubed, which equals 4."},{"Start":"00:48.420 ","End":"00:50.985","Text":"Minus x is minus 1 times x,"},{"Start":"00:50.985 ","End":"00:53.165","Text":"so we get minus 1 cubed,"},{"Start":"00:53.165 ","End":"00:58.680","Text":"x cubed, and this equals minus 4 x cubed."},{"Start":"00:58.680 ","End":"01:05.000","Text":"Notice that for x cubed is simply f of x so we get minus f of x."},{"Start":"01:05.000 ","End":"01:11.105","Text":"This means that f of minus x is equal to minus f of x,"},{"Start":"01:11.105 ","End":"01:13.070","Text":"making the function odd."},{"Start":"01:13.070 ","End":"01:16.600","Text":"We\u0027re done."}],"ID":4356},{"Watched":false,"Name":"Exercise 2","Duration":"1m 15s","ChapterTopicVideoID":4348,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we\u0027re given 2 functions,"},{"Start":"00:03.060 ","End":"00:05.730","Text":"f of x and g of x as described."},{"Start":"00:05.730 ","End":"00:10.920","Text":"We have to compute what is the value of g composed f of 1."},{"Start":"00:10.920 ","End":"00:14.010","Text":"Let\u0027s recall what g composed f means."},{"Start":"00:14.010 ","End":"00:21.285","Text":"In general, g composed f of a value x is just g of f of x,"},{"Start":"00:21.285 ","End":"00:25.455","Text":"which means we first compute f of x and then apply g to the answer of that."},{"Start":"00:25.455 ","End":"00:29.955","Text":"In our case, we have g composed f of 1,"},{"Start":"00:29.955 ","End":"00:33.645","Text":"and that means g of f of 1."},{"Start":"00:33.645 ","End":"00:38.240","Text":"I omitted the brackets better to leave them in but not mandatory."},{"Start":"00:38.240 ","End":"00:42.335","Text":"First of all, let\u0027s compute what is f of 1,"},{"Start":"00:42.335 ","End":"00:45.275","Text":"and I\u0027ll do this at the side here."},{"Start":"00:45.275 ","End":"00:49.460","Text":"F of 1, looking at the definition of f of x,"},{"Start":"00:49.460 ","End":"00:51.110","Text":"which is x minus 4,"},{"Start":"00:51.110 ","End":"00:53.555","Text":"is equal to 1 minus 4,"},{"Start":"00:53.555 ","End":"00:55.790","Text":"which is minus 3."},{"Start":"00:55.790 ","End":"00:59.740","Text":"This minus 3 is what I have to substitute here."},{"Start":"00:59.740 ","End":"01:04.470","Text":"I get g of this same minus 3 from here."},{"Start":"01:04.470 ","End":"01:08.690","Text":"Now I look at the definition of g. G of x is x squared."},{"Start":"01:08.690 ","End":"01:11.495","Text":"So I have minus 3 squared,"},{"Start":"01:11.495 ","End":"01:13.970","Text":"which is equal to 9."},{"Start":"01:13.970 ","End":"01:16.350","Text":"That\u0027s our answer."}],"ID":4357},{"Watched":false,"Name":"Exercise 3","Duration":"51s","ChapterTopicVideoID":4349,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.325","Text":"In this exercise, we\u0027re given 2 functions,"},{"Start":"00:02.325 ","End":"00:04.545","Text":"f and g as described."},{"Start":"00:04.545 ","End":"00:07.560","Text":"We have to compute their composition f compose"},{"Start":"00:07.560 ","End":"00:12.390","Text":"g. Remember that f compose g when applied to x,"},{"Start":"00:12.390 ","End":"00:16.365","Text":"means f of g of x."},{"Start":"00:16.365 ","End":"00:21.840","Text":"Which means that we first apply g to x and then f to the answer of that."},{"Start":"00:21.840 ","End":"00:26.040","Text":"Let\u0027s start with evaluating what is g of x?"},{"Start":"00:26.040 ","End":"00:28.770","Text":"That is simply, remember, g of x,"},{"Start":"00:28.770 ","End":"00:31.710","Text":"I\u0027m just copying it from above is x squared."},{"Start":"00:31.710 ","End":"00:36.645","Text":"I want to substitute x squared where I had g of x."},{"Start":"00:36.645 ","End":"00:39.830","Text":"We get f of x squared."},{"Start":"00:39.830 ","End":"00:44.040","Text":"Now I have to use the definition of f. F of x is x minus 4."},{"Start":"00:44.040 ","End":"00:50.205","Text":"I put x squared in place of x and I get x squared minus 4."},{"Start":"00:50.205 ","End":"00:52.540","Text":"That\u0027s our answer."}],"ID":4358},{"Watched":false,"Name":"Exercise 4","Duration":"53s","ChapterTopicVideoID":4350,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this exercise, we\u0027re given 2 functions,"},{"Start":"00:02.220 ","End":"00:07.980","Text":"f and g as described and we have to compute what is g composed f of x."},{"Start":"00:07.980 ","End":"00:11.475","Text":"Now, g composed f of x, as you recall,"},{"Start":"00:11.475 ","End":"00:14.985","Text":"means g of f of x,"},{"Start":"00:14.985 ","End":"00:19.350","Text":"which means you first apply f to x and then g to the answer."},{"Start":"00:19.350 ","End":"00:25.065","Text":"Now, f of x is written above here as x minus 4."},{"Start":"00:25.065 ","End":"00:30.015","Text":"So what we have to do is instead of f of x to put x minus 4."},{"Start":"00:30.015 ","End":"00:31.890","Text":"Now we have g of something."},{"Start":"00:31.890 ","End":"00:38.500","Text":"g of x is x squared so g of x minus 4 is x minus 4 squared,"},{"Start":"00:38.500 ","End":"00:42.740","Text":"and we could leave the answer like that or we could expand it using 1 of"},{"Start":"00:42.740 ","End":"00:48.305","Text":"the formulae and get x squared minus 8x plus 16."},{"Start":"00:48.305 ","End":"00:49.910","Text":"That\u0027s our answer. But of course,"},{"Start":"00:49.910 ","End":"00:53.970","Text":"you could leave it like this also. We\u0027re done."}],"ID":4359},{"Watched":false,"Name":"Exercise 5","Duration":"59s","ChapterTopicVideoID":4351,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this exercise, we\u0027re given a function f,"},{"Start":"00:02.610 ","End":"00:05.745","Text":"where f of x is x minus 4 and we have to compute"},{"Start":"00:05.745 ","End":"00:09.525","Text":"what is the new function f composed f of x."},{"Start":"00:09.525 ","End":"00:11.640","Text":"Normally one sees f composed with g,"},{"Start":"00:11.640 ","End":"00:14.820","Text":"but there\u0027s no reason why you can\u0027t compose a function with itself."},{"Start":"00:14.820 ","End":"00:16.680","Text":"Now let\u0027s recall what this means."},{"Start":"00:16.680 ","End":"00:24.255","Text":"F composed f of x means f. This f is this f of f,"},{"Start":"00:24.255 ","End":"00:26.625","Text":"which is the right f of x."},{"Start":"00:26.625 ","End":"00:32.775","Text":"Which means that we first apply f to x and then we apply f again to the answer."},{"Start":"00:32.775 ","End":"00:38.770","Text":"Leaving the outer f alone inside f of x is x minus 4,"},{"Start":"00:38.770 ","End":"00:42.440","Text":"so we have f of x minus 4."},{"Start":"00:42.440 ","End":"00:44.825","Text":"Now compute f of x minus 4,"},{"Start":"00:44.825 ","End":"00:48.140","Text":"we just put x minus 4 instead of x,"},{"Start":"00:48.140 ","End":"00:53.645","Text":"so we get x minus 4 instead of x minus 4."},{"Start":"00:53.645 ","End":"01:00.300","Text":"In other words, the answer is x minus 8, and we\u0027re done."}],"ID":4360},{"Watched":false,"Name":"Exercise 6","Duration":"1m 42s","ChapterTopicVideoID":4352,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"This exercise involves the composition of 3 different functions,"},{"Start":"00:03.900 ","End":"00:08.490","Text":"f, g, and h, each described accordingly."},{"Start":"00:08.490 ","End":"00:13.995","Text":"We have to compute what is h of g of f of 5."},{"Start":"00:13.995 ","End":"00:16.920","Text":"The way to do this is to work from the inside out."},{"Start":"00:16.920 ","End":"00:19.245","Text":"In other words, first to apply f to 5,"},{"Start":"00:19.245 ","End":"00:21.150","Text":"then g to the answer of that,"},{"Start":"00:21.150 ","End":"00:24.935","Text":"and then h to the answer of that and I\u0027ll show you what I mean."},{"Start":"00:24.935 ","End":"00:30.380","Text":"H of g of f of 5 is equal to ,"},{"Start":"00:30.380 ","End":"00:34.295","Text":"the first thing I\u0027m going to do is compute f of 5."},{"Start":"00:34.295 ","End":"00:37.550","Text":"That is equal to looking at the definition above,"},{"Start":"00:37.550 ","End":"00:40.820","Text":"x minus 4 when x is 5 is simply 1."},{"Start":"00:40.820 ","End":"00:44.210","Text":"I do this as an exercise at the side and say,"},{"Start":"00:44.210 ","End":"00:48.755","Text":"f of 5 is 5 minus 4,"},{"Start":"00:48.755 ","End":"00:51.490","Text":"which is equal to 1."},{"Start":"00:51.490 ","End":"00:55.620","Text":"This equals h of g of,"},{"Start":"00:55.620 ","End":"00:59.085","Text":"I simply take this 1 here and put it 1."},{"Start":"00:59.085 ","End":"01:02.765","Text":"Again, working from the inside out."},{"Start":"01:02.765 ","End":"01:05.605","Text":"Next thing I\u0027m going to compute is g of 1,"},{"Start":"01:05.605 ","End":"01:09.060","Text":"and I\u0027ll do that at the side also and say,"},{"Start":"01:09.060 ","End":"01:11.400","Text":"g of 1 is equal to;"},{"Start":"01:11.400 ","End":"01:17.415","Text":"g of x is x squared so g of 1 is 1 squared, which is 1."},{"Start":"01:17.415 ","End":"01:18.825","Text":"Now I\u0027m continuing."},{"Start":"01:18.825 ","End":"01:20.550","Text":"This is h of;"},{"Start":"01:20.550 ","End":"01:23.550","Text":"now g of 1 has being computed and here it is."},{"Start":"01:23.550 ","End":"01:27.295","Text":"I simply put this in here and I have h of 1."},{"Start":"01:27.295 ","End":"01:29.610","Text":"Now I refer to the definition of h,"},{"Start":"01:29.610 ","End":"01:31.695","Text":"h of x is 4 over x."},{"Start":"01:31.695 ","End":"01:33.885","Text":"There\u0027s no need to do it at the side anymore,"},{"Start":"01:33.885 ","End":"01:36.810","Text":"it\u0027s equal to 4 over 1,"},{"Start":"01:36.810 ","End":"01:38.165","Text":"x is 1 in our case,"},{"Start":"01:38.165 ","End":"01:40.160","Text":"and this is equal to 4."},{"Start":"01:40.160 ","End":"01:42.690","Text":"That\u0027s our answer. We\u0027re done."}],"ID":4361},{"Watched":false,"Name":"Exercise 7","Duration":"1m 45s","ChapterTopicVideoID":4366,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"In this exercise, we\u0027re given 3 functions;"},{"Start":"00:02.805 ","End":"00:05.220","Text":"f of x, g of x,"},{"Start":"00:05.220 ","End":"00:07.680","Text":"and h of x as described,"},{"Start":"00:07.680 ","End":"00:11.055","Text":"we have to compute seemingly complicated expression,"},{"Start":"00:11.055 ","End":"00:13.125","Text":"h of g of f of x."},{"Start":"00:13.125 ","End":"00:15.195","Text":"But really this is quite simple."},{"Start":"00:15.195 ","End":"00:17.400","Text":"The idea is to work from the inside out,"},{"Start":"00:17.400 ","End":"00:18.855","Text":"compute f of x,"},{"Start":"00:18.855 ","End":"00:20.790","Text":"apply g to the answer of that,"},{"Start":"00:20.790 ","End":"00:22.605","Text":"and apply h to the answer to that."},{"Start":"00:22.605 ","End":"00:24.450","Text":"I\u0027ll show you in detail what I mean."},{"Start":"00:24.450 ","End":"00:26.340","Text":"We\u0027re working inside out."},{"Start":"00:26.340 ","End":"00:29.970","Text":"The first thing we\u0027re going to do is to compute f of x,"},{"Start":"00:29.970 ","End":"00:31.905","Text":"which is not really a computation,"},{"Start":"00:31.905 ","End":"00:35.070","Text":"we simply have to copy it from the definition."},{"Start":"00:35.070 ","End":"00:37.320","Text":"f of x is x minus 4."},{"Start":"00:37.320 ","End":"00:46.300","Text":"The x minus 4, I\u0027m going to put in place of f of x and get h of g of x minus 4."},{"Start":"00:46.300 ","End":"00:49.010","Text":"Now, again, going from inside out,"},{"Start":"00:49.010 ","End":"00:56.105","Text":"the next thing is to compute g of x minus 4. g is given by g of x is x squared."},{"Start":"00:56.105 ","End":"00:58.955","Text":"In place of x, I have to put x minus 4."},{"Start":"00:58.955 ","End":"01:08.300","Text":"This I could do at the side and get g of x minus 4 is equal to x minus 4 squared."},{"Start":"01:08.300 ","End":"01:09.650","Text":"I\u0027ll leave it in that form,"},{"Start":"01:09.650 ","End":"01:12.560","Text":"if I need to expand it later, might do it."},{"Start":"01:12.560 ","End":"01:19.640","Text":"We get that this is equal to h of x minus 4 squared,"},{"Start":"01:19.640 ","End":"01:22.835","Text":"which is simply what happens when I this here."},{"Start":"01:22.835 ","End":"01:28.970","Text":"The last thing that we need to do is to compute h of x minus 4 squared."},{"Start":"01:28.970 ","End":"01:30.360","Text":"Looking at h of x,"},{"Start":"01:30.360 ","End":"01:32.655","Text":"h of x is 4 over x."},{"Start":"01:32.655 ","End":"01:34.700","Text":"Instead of x, we put this whole thing,"},{"Start":"01:34.700 ","End":"01:36.230","Text":"x minus 4 squared."},{"Start":"01:36.230 ","End":"01:43.250","Text":"The answer is 4 over x minus 4 squared and that\u0027s it."},{"Start":"01:43.250 ","End":"01:46.020","Text":"This is the answer, and we\u0027re done."}],"ID":4375},{"Watched":false,"Name":"Exercise 8","Duration":"49s","ChapterTopicVideoID":4367,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.925","Text":"In this exercise, we\u0027re given 3 functions,"},{"Start":"00:02.925 ","End":"00:06.914","Text":"f, g, and h, each appropriately described."},{"Start":"00:06.914 ","End":"00:10.694","Text":"We have to compute what is h of g of x."},{"Start":"00:10.694 ","End":"00:14.910","Text":"Notice that we don\u0027t actually need the value of f in this exercise,"},{"Start":"00:14.910 ","End":"00:21.635","Text":"all we need is h and g. H of g of x is equal to,"},{"Start":"00:21.635 ","End":"00:27.825","Text":"the first thing to do is to replace the g of x here with what it actually is."},{"Start":"00:27.825 ","End":"00:31.215","Text":"We look here, g of x is x squared."},{"Start":"00:31.215 ","End":"00:35.254","Text":"What we get is h of x squared."},{"Start":"00:35.254 ","End":"00:37.685","Text":"Now, the definition of h is here,"},{"Start":"00:37.685 ","End":"00:39.110","Text":"h of x is 4/x."},{"Start":"00:39.110 ","End":"00:41.765","Text":"What we do is we put x squared in place of x,"},{"Start":"00:41.765 ","End":"00:46.830","Text":"so we simply get 4/x squared."},{"Start":"00:46.830 ","End":"00:49.570","Text":"That\u0027s the answer. We\u0027re done."}],"ID":4376},{"Watched":false,"Name":"Exercise 9","Duration":"4m 8s","ChapterTopicVideoID":4368,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"In this exercise, we\u0027re given 2 functions, f of x,"},{"Start":"00:03.330 ","End":"00:04.875","Text":"which is x squared plus 4,"},{"Start":"00:04.875 ","End":"00:08.265","Text":"and g of x, which is the square root of x minus 4,"},{"Start":"00:08.265 ","End":"00:10.410","Text":"and there are 3 separate questions a,"},{"Start":"00:10.410 ","End":"00:12.405","Text":"b, and c. In a,"},{"Start":"00:12.405 ","End":"00:19.890","Text":"we\u0027re asked to compute what is f compose g of x. F composed g of x,"},{"Start":"00:19.890 ","End":"00:25.245","Text":"if you recall, means f of g of x."},{"Start":"00:25.245 ","End":"00:28.855","Text":"Other words, apply g to x and f to the answer of that."},{"Start":"00:28.855 ","End":"00:32.285","Text":"We\u0027re going to compute first of all, g of x,"},{"Start":"00:32.285 ","End":"00:36.185","Text":"or rather we don\u0027t have to compute it because it\u0027s written above."},{"Start":"00:36.185 ","End":"00:41.440","Text":"This is f of the square root of x minus 4."},{"Start":"00:41.440 ","End":"00:46.760","Text":"Now, we have to look at the definition of f. F of x is x squared plus 4."},{"Start":"00:46.760 ","End":"00:56.305","Text":"We put this expression instead of x and get square root of x minus 4 squared plus 4."},{"Start":"00:56.305 ","End":"01:01.700","Text":"This equals, square root of anything squared gives us the thing itself"},{"Start":"01:01.700 ","End":"01:08.165","Text":"which is x minus 4 plus the 4 from here which equals x."},{"Start":"01:08.165 ","End":"01:13.955","Text":"In other words, the answer of f compose g of x is just x itself."},{"Start":"01:13.955 ","End":"01:18.110","Text":"We\u0027re going to save this result before we move on to part B."},{"Start":"01:18.110 ","End":"01:22.130","Text":"We have to find what is the domain of f compose g of x."},{"Start":"01:22.130 ","End":"01:23.555","Text":"Now, at first sight,"},{"Start":"01:23.555 ","End":"01:26.060","Text":"you would look here the answer of a and say,"},{"Start":"01:26.060 ","End":"01:28.700","Text":"well, since this function is equal to x,"},{"Start":"01:28.700 ","End":"01:30.710","Text":"x is defined everywhere."},{"Start":"01:30.710 ","End":"01:33.125","Text":"The domain would be all x."},{"Start":"01:33.125 ","End":"01:34.895","Text":"But that\u0027s not quite right."},{"Start":"01:34.895 ","End":"01:39.080","Text":"The reason it\u0027s not quite right is because we got there by first"},{"Start":"01:39.080 ","End":"01:43.535","Text":"applying g and then applying f. In other words,"},{"Start":"01:43.535 ","End":"01:51.920","Text":"because f composed g of x is equal to f of g of x,"},{"Start":"01:51.920 ","End":"01:55.920","Text":"we have to make sure that g of x is defined, i.e."},{"Start":"01:55.920 ","End":"01:59.345","Text":"that x is in the domain of g. Now, looking at g,"},{"Start":"01:59.345 ","End":"02:03.680","Text":"g of x equals the square root of x minus 4."},{"Start":"02:03.680 ","End":"02:05.230","Text":"If this is not defined,"},{"Start":"02:05.230 ","End":"02:07.160","Text":"then we can\u0027t begin the process."},{"Start":"02:07.160 ","End":"02:09.245","Text":"When is this defined?"},{"Start":"02:09.245 ","End":"02:12.575","Text":"The square root is defined for all non-negative numbers."},{"Start":"02:12.575 ","End":"02:17.900","Text":"This requires that x minus 4 be non-negative, i.e."},{"Start":"02:17.900 ","End":"02:19.565","Text":"bigger or equal to 0,"},{"Start":"02:19.565 ","End":"02:20.870","Text":"or in other words,"},{"Start":"02:20.870 ","End":"02:23.990","Text":"that x is bigger or equal to 4."},{"Start":"02:23.990 ","End":"02:25.490","Text":"That\u0027s our answer."},{"Start":"02:25.490 ","End":"02:27.190","Text":"We\u0027re done for part B."},{"Start":"02:27.190 ","End":"02:34.570","Text":"Let\u0027s move on to part C. Part C asks what is the domain of g composed f of x?"},{"Start":"02:34.570 ","End":"02:36.040","Text":"Just like in part A,"},{"Start":"02:36.040 ","End":"02:38.210","Text":"let\u0027s do the computation first."},{"Start":"02:38.210 ","End":"02:41.285","Text":"G of f of x,"},{"Start":"02:41.285 ","End":"02:44.075","Text":"which is what g composed f of x means,"},{"Start":"02:44.075 ","End":"02:46.850","Text":"is equal to g of."},{"Start":"02:46.850 ","End":"02:51.005","Text":"Now, f of x should be replaced with its definition,"},{"Start":"02:51.005 ","End":"02:52.380","Text":"x squared plus 4,"},{"Start":"02:52.380 ","End":"02:56.415","Text":"so we get g of x squared plus 4."},{"Start":"02:56.415 ","End":"02:59.510","Text":"Now, looking at the definition of g of x,"},{"Start":"02:59.510 ","End":"03:01.900","Text":"g of x is the square root of x minus 4,"},{"Start":"03:01.900 ","End":"03:03.695","Text":"so we just instead of x,"},{"Start":"03:03.695 ","End":"03:05.150","Text":"put this whole expression."},{"Start":"03:05.150 ","End":"03:09.080","Text":"We get the square root of instead of x,"},{"Start":"03:09.080 ","End":"03:13.220","Text":"x squared plus 4 minus the 4 that was there,"},{"Start":"03:13.220 ","End":"03:17.260","Text":"which equals the square root of x squared."},{"Start":"03:17.260 ","End":"03:22.745","Text":"That\u0027s the computation of g composed f. In other words,"},{"Start":"03:22.745 ","End":"03:30.065","Text":"g composed f of x is equal to the square root of x squared."},{"Start":"03:30.065 ","End":"03:32.615","Text":"Now, we were asked about the domain."},{"Start":"03:32.615 ","End":"03:34.280","Text":"Looking at this thing,"},{"Start":"03:34.280 ","End":"03:35.550","Text":"x could be anything."},{"Start":"03:35.550 ","End":"03:37.235","Text":"Because whatever you put for x,"},{"Start":"03:37.235 ","End":"03:38.420","Text":"positive or negative,"},{"Start":"03:38.420 ","End":"03:42.565","Text":"x squared is going to be non-negative and so it\u0027s defined."},{"Start":"03:42.565 ","End":"03:44.580","Text":"We would say all x,"},{"Start":"03:44.580 ","End":"03:47.870","Text":"but remembering the trouble we had in part B,"},{"Start":"03:47.870 ","End":"03:52.175","Text":"we have to look and make sure that f of x is"},{"Start":"03:52.175 ","End":"03:57.580","Text":"defined for all x. F of x was simply x squared plus 4."},{"Start":"03:57.580 ","End":"03:59.615","Text":"There\u0027s no problem here."},{"Start":"03:59.615 ","End":"04:08.890","Text":"In this case, we can safely say that the domain of definition is all x. We\u0027re done."}],"ID":4377},{"Watched":false,"Name":"Exercise 10","Duration":"5m 32s","ChapterTopicVideoID":4369,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this exercise, we\u0027re given 2 functions,"},{"Start":"00:02.820 ","End":"00:05.280","Text":"f of x and g of x as follows."},{"Start":"00:05.280 ","End":"00:08.800","Text":"Notice that g of x is defined in terms of some constant k,"},{"Start":"00:08.800 ","End":"00:10.500","Text":"and we\u0027re going to have to find the value of"},{"Start":"00:10.500 ","End":"00:14.565","Text":"the constant k for which f of g of x is equal to x."},{"Start":"00:14.565 ","End":"00:17.400","Text":"To be precise, it has to be equal to x for all x,"},{"Start":"00:17.400 ","End":"00:22.530","Text":"and sometimes this is written as f of g of x"},{"Start":"00:22.530 ","End":"00:25.815","Text":"is identically equal to x,"},{"Start":"00:25.815 ","End":"00:28.065","Text":"that\u0027s the for all x part."},{"Start":"00:28.065 ","End":"00:31.340","Text":"We begin by computing f of g of x."},{"Start":"00:31.340 ","End":"00:38.240","Text":"F of g of x is equal to f of,"},{"Start":"00:38.240 ","End":"00:40.115","Text":"we substitute g of x,"},{"Start":"00:40.115 ","End":"00:46.125","Text":"which is 1 minus 2x over x plus k,"},{"Start":"00:46.125 ","End":"00:49.070","Text":"and what we have to do to compute this"},{"Start":"00:49.070 ","End":"00:51.230","Text":"is to substitute this whole fraction"},{"Start":"00:51.230 ","End":"00:55.725","Text":"instead of x, both here and here."},{"Start":"00:55.725 ","End":"01:05.939","Text":"What we get is 1 minus 2x over x plus k,"},{"Start":"01:05.939 ","End":"01:07.560","Text":"that\u0027s this x here,"},{"Start":"01:07.560 ","End":"01:12.525","Text":"plus 1 all over 1 minus"},{"Start":"01:12.525 ","End":"01:19.575","Text":"2x over x plus k plus 2 from here."},{"Start":"01:19.575 ","End":"01:22.395","Text":"This needs to be simplified."},{"Start":"01:22.395 ","End":"01:24.880","Text":"Let\u0027s put a common denominator"},{"Start":"01:24.880 ","End":"01:28.430","Text":"in both the numerator and the denominator of this."},{"Start":"01:28.430 ","End":"01:30.315","Text":"This is equal to,"},{"Start":"01:30.315 ","End":"01:34.745","Text":"here we have 1 minus 2x."},{"Start":"01:34.745 ","End":"01:39.210","Text":"The whole thing is going to have to be over x plus k,"},{"Start":"01:39.210 ","End":"01:41.890","Text":"and if you know your fractions,"},{"Start":"01:41.890 ","End":"01:45.340","Text":"we have to multiply this 1 top and bottom by x plus k."},{"Start":"01:45.340 ","End":"01:48.760","Text":"We get x plus k."},{"Start":"01:48.760 ","End":"01:51.115","Text":"All this is the numerator,"},{"Start":"01:51.115 ","End":"01:53.485","Text":"on the denominator, something similar."},{"Start":"01:53.485 ","End":"01:57.080","Text":"We have something over x plus k,"},{"Start":"01:57.080 ","End":"02:00.525","Text":"and again the same 1 minus 2x from here,"},{"Start":"02:00.525 ","End":"02:04.180","Text":"and this time plus twice x plus k."},{"Start":"02:04.180 ","End":"02:06.370","Text":"Let\u0027s simplify this further."},{"Start":"02:06.370 ","End":"02:08.155","Text":"It looks quite a mess."},{"Start":"02:08.155 ","End":"02:09.130","Text":"What we\u0027ll do is,"},{"Start":"02:09.130 ","End":"02:11.770","Text":"since we have a numerator and denominator,"},{"Start":"02:11.770 ","End":"02:14.890","Text":"both of which are fractions over x plus k."},{"Start":"02:14.890 ","End":"02:18.490","Text":"We can actually just simply as if to cancel"},{"Start":"02:18.490 ","End":"02:21.205","Text":"the x plus k from here and here,"},{"Start":"02:21.205 ","End":"02:24.130","Text":"and get a simple fraction which is"},{"Start":"02:24.130 ","End":"02:33.945","Text":"1 minus 2x plus x plus k over 1 minus 2x,"},{"Start":"02:33.945 ","End":"02:35.775","Text":"open the brackets here,"},{"Start":"02:35.775 ","End":"02:39.325","Text":"plus 2x plus 2k."},{"Start":"02:39.325 ","End":"02:42.140","Text":"Let\u0027s continue simplifying."},{"Start":"02:42.140 ","End":"02:45.290","Text":"We\u0027ll collect the x\u0027s together in the top,"},{"Start":"02:45.290 ","End":"02:52.095","Text":"and we get minus x plus k plus 1 over,"},{"Start":"02:52.095 ","End":"02:55.590","Text":"we\u0027re lucky minus 2x and plus 2x cancel,"},{"Start":"02:55.590 ","End":"03:00.710","Text":"and all we get in the denominator is 2k plus 1."},{"Start":"03:00.710 ","End":"03:02.750","Text":"This whole expression here"},{"Start":"03:02.750 ","End":"03:07.205","Text":"is equal to the original f of g of x."},{"Start":"03:07.205 ","End":"03:12.350","Text":"Now, we\u0027re told in the exercise that we want f of"},{"Start":"03:12.350 ","End":"03:18.785","Text":"g of x to be equal or let\u0027s say identically equal to x."},{"Start":"03:18.785 ","End":"03:25.090","Text":"In other words, this thing has to be identically equal to x."},{"Start":"03:25.090 ","End":"03:30.799","Text":"Minus x plus k plus 1"},{"Start":"03:30.799 ","End":"03:37.720","Text":"over 2k plus 1 is identically equal to x."},{"Start":"03:37.720 ","End":"03:39.995","Text":"The easiest way to do this,"},{"Start":"03:39.995 ","End":"03:42.260","Text":"because we have an identity,"},{"Start":"03:42.260 ","End":"03:44.765","Text":"x can be any value we want."},{"Start":"03:44.765 ","End":"03:46.220","Text":"The easiest thing to do would be"},{"Start":"03:46.220 ","End":"03:48.140","Text":"to substitute x equals 0"},{"Start":"03:48.140 ","End":"03:49.790","Text":"and get an equation in k."},{"Start":"03:49.790 ","End":"03:50.860","Text":"Let\u0027s try that."},{"Start":"03:50.860 ","End":"03:53.240","Text":"In this equation, we\u0027ll put x equals 0"},{"Start":"03:53.240 ","End":"03:55.145","Text":"since it\u0027s true for all x,"},{"Start":"03:55.145 ","End":"03:57.710","Text":"and we get x is 0."},{"Start":"03:57.710 ","End":"04:01.250","Text":"We\u0027re simply left with k plus 1"},{"Start":"04:01.250 ","End":"04:07.339","Text":"over 2k plus 1 is equal to 0."},{"Start":"04:07.339 ","End":"04:09.530","Text":"Now we have a fraction equal to 0,"},{"Start":"04:09.530 ","End":"04:12.020","Text":"and whenever a fraction is equal to 0,"},{"Start":"04:12.020 ","End":"04:14.200","Text":"the numerator has to be 0,"},{"Start":"04:14.200 ","End":"04:18.915","Text":"so we get that k plus 1 equals 0."},{"Start":"04:18.915 ","End":"04:24.145","Text":"In other words, k is equal to minus 1."},{"Start":"04:24.145 ","End":"04:26.075","Text":"Now, that\u0027s the answer,"},{"Start":"04:26.075 ","End":"04:28.160","Text":"but our work is not quite done"},{"Start":"04:28.160 ","End":"04:32.390","Text":"because we know that this holds true for x equals 0."},{"Start":"04:32.390 ","End":"04:35.570","Text":"But we have to make sure it\u0027s true for all x."},{"Start":"04:35.570 ","End":"04:37.580","Text":"What we basically have to do"},{"Start":"04:37.580 ","End":"04:42.230","Text":"is put k equals minus 1 into this expression"},{"Start":"04:42.230 ","End":"04:46.700","Text":"and see that we really do get exactly x for all x."},{"Start":"04:46.700 ","End":"04:50.090","Text":"Putting k equals minus 1 in there,"},{"Start":"04:50.090 ","End":"04:55.835","Text":"we get that f of g of x was equal to,"},{"Start":"04:55.835 ","End":"04:57.800","Text":"now k is minus 1,"},{"Start":"04:57.800 ","End":"05:03.540","Text":"so we get minus x minus 1 plus 1"},{"Start":"05:03.540 ","End":"05:07.480","Text":"over twice minus 1 plus 1."},{"Start":"05:07.480 ","End":"05:11.255","Text":"This equals minus 1 plus 1 cancel,"},{"Start":"05:11.255 ","End":"05:14.135","Text":"and twice minus 1 plus 1 is minus 1."},{"Start":"05:14.135 ","End":"05:18.375","Text":"It\u0027s minus x over minus 1."},{"Start":"05:18.375 ","End":"05:22.070","Text":"In other words, it\u0027s just equal to x,"},{"Start":"05:22.070 ","End":"05:23.840","Text":"and this was true for all x."},{"Start":"05:23.840 ","End":"05:29.540","Text":"So we indeed get that f of g of x is equal to x for all x,"},{"Start":"05:29.540 ","End":"05:33.150","Text":"could have put the third line here, and we\u0027re done."}],"ID":4378},{"Watched":false,"Name":"Exercise 11","Duration":"3m 39s","ChapterTopicVideoID":4370,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this exercise, we\u0027re given 2 functions,"},{"Start":"00:02.370 ","End":"00:04.919","Text":"f and g, defined on the set of real numbers."},{"Start":"00:04.919 ","End":"00:07.860","Text":"G of x is given by ax minus 3,"},{"Start":"00:07.860 ","End":"00:09.630","Text":"and f of x is ax plus 5,"},{"Start":"00:09.630 ","End":"00:11.745","Text":"where a is some constant."},{"Start":"00:11.745 ","End":"00:14.670","Text":"Now, we\u0027re given that f compose g of x"},{"Start":"00:14.670 ","End":"00:18.255","Text":"is the same as g composed f of x for all x."},{"Start":"00:18.255 ","End":"00:21.300","Text":"We have to find out what is the value of a."},{"Start":"00:21.300 ","End":"00:23.965","Text":"We have a left-hand side and a right-hand side,"},{"Start":"00:23.965 ","End":"00:26.280","Text":"and let\u0027s compute each 1 separately."},{"Start":"00:26.280 ","End":"00:29.580","Text":"We\u0027ll begin with f compose g of x,"},{"Start":"00:29.580 ","End":"00:35.010","Text":"and that is equal to f of g of x,"},{"Start":"00:35.010 ","End":"00:37.455","Text":"which is equal to f of,"},{"Start":"00:37.455 ","End":"00:41.775","Text":"we substitute g of x from here, ax minus 3."},{"Start":"00:41.775 ","End":"00:47.450","Text":"Now we substitute ax minus 3 into the definition of f of x in place of x"},{"Start":"00:47.450 ","End":"00:54.890","Text":"and we get a times ax minus 3 plus 5."},{"Start":"00:54.890 ","End":"00:56.390","Text":"We could simplify it,"},{"Start":"00:56.390 ","End":"00:59.960","Text":"but I think for the time being we\u0027ll just leave it as it is."},{"Start":"00:59.960 ","End":"01:02.855","Text":"Next, we\u0027ll move on to the right-hand side,"},{"Start":"01:02.855 ","End":"01:05.630","Text":"g composed f of x."},{"Start":"01:05.630 ","End":"01:10.880","Text":"This is equal to g of f of x,"},{"Start":"01:10.880 ","End":"01:13.675","Text":"which is equal to g of,"},{"Start":"01:13.675 ","End":"01:16.460","Text":"f of x is ax plus 5."},{"Start":"01:16.460 ","End":"01:18.175","Text":"This is equal to,"},{"Start":"01:18.175 ","End":"01:22.189","Text":"looking at the definition of g and substituting x,"},{"Start":"01:22.189 ","End":"01:29.740","Text":"we get a times ax plus 5 minus 3."},{"Start":"01:29.740 ","End":"01:32.389","Text":"That is our right-hand side."},{"Start":"01:32.389 ","End":"01:37.135","Text":"Now we have to compare the left-hand side to the right-hand side."},{"Start":"01:37.135 ","End":"01:43.010","Text":"Let\u0027s write a ax minus 3 plus 5."},{"Start":"01:43.010 ","End":"01:44.780","Text":"I\u0027m just copying the left-hand side"},{"Start":"01:44.780 ","End":"01:47.480","Text":"and equating it to the right-hand side,"},{"Start":"01:47.480 ","End":"01:51.975","Text":"a ax plus 5 minus 3."},{"Start":"01:51.975 ","End":"01:56.480","Text":"Now 1 easy way to proceed is that since this is true for all x,"},{"Start":"01:56.480 ","End":"01:59.345","Text":"we can substitute any value of x we want."},{"Start":"01:59.345 ","End":"02:07.230","Text":"For example, we could try substituting x equals 0."},{"Start":"02:07.230 ","End":"02:14.440","Text":"This would give us a times minus 3 plus 5 equals a,"},{"Start":"02:14.440 ","End":"02:16.450","Text":"our x is 0 so this is 0,"},{"Start":"02:16.450 ","End":"02:19.475","Text":"times 5 minus 3."},{"Start":"02:19.475 ","End":"02:27.205","Text":"In other words, minus 3a plus 5 equals 5a minus 3."},{"Start":"02:27.205 ","End":"02:30.454","Text":"Bring the a\u0027s over to the right-hand side,"},{"Start":"02:30.454 ","End":"02:32.285","Text":"and we\u0027ve got 8a."},{"Start":"02:32.285 ","End":"02:35.120","Text":"Bring the 3 over to the left-hand side,"},{"Start":"02:35.120 ","End":"02:36.560","Text":"and we\u0027ve got 8."},{"Start":"02:36.560 ","End":"02:40.835","Text":"So a is equal to 1."},{"Start":"02:40.835 ","End":"02:44.990","Text":"I would say that we\u0027re finished except that truthfully what we have to"},{"Start":"02:44.990 ","End":"02:49.520","Text":"do is to substitute a and make sure that this is true for all x,"},{"Start":"02:49.520 ","End":"02:53.180","Text":"because as we know now it\u0027s just true for x equals 0."},{"Start":"02:53.180 ","End":"02:55.445","Text":"Let\u0027s substitute a equals 1,"},{"Start":"02:55.445 ","End":"02:57.425","Text":"both here and here."},{"Start":"02:57.425 ","End":"03:05.460","Text":"We got 1 times 1x minus 3 plus 5; is it equal?"},{"Start":"03:05.460 ","End":"03:13.790","Text":"We have to check that it\u0027s always equal to 1 times 1x plus 5 minus 3."},{"Start":"03:13.790 ","End":"03:19.205","Text":"Now this side, if we look at it is just x minus 3 plus 5,"},{"Start":"03:19.205 ","End":"03:21.955","Text":"so it\u0027s x plus 2."},{"Start":"03:21.955 ","End":"03:24.840","Text":"Here we have 1 times 1 x plus 5,"},{"Start":"03:24.840 ","End":"03:27.290","Text":"is just x plus 5 minus 3."},{"Start":"03:27.290 ","End":"03:31.000","Text":"This side, it also equals x plus 2."},{"Start":"03:31.000 ","End":"03:33.485","Text":"This is equal always for all x,"},{"Start":"03:33.485 ","End":"03:36.590","Text":"sometimes written as 3 lines, it\u0027s always equal."},{"Start":"03:36.590 ","End":"03:39.630","Text":"This time we really are done."}],"ID":4379},{"Watched":false,"Name":"Exercise 12","Duration":"1m 53s","ChapterTopicVideoID":4371,"CourseChapterTopicPlaylistID":1182,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have to compute the composition f"},{"Start":"00:03.720 ","End":"00:08.535","Text":"compose g of 2 functions which are both defined piecewise."},{"Start":"00:08.535 ","End":"00:13.020","Text":"F of x is defined as 4x plus 3 for x less than 5,"},{"Start":"00:13.020 ","End":"00:15.795","Text":"but 2x for x bigger or equal to 5."},{"Start":"00:15.795 ","End":"00:22.455","Text":"Similarly, g is defined piecewise as 3 when x is bigger or equal to 1 and 2,"},{"Start":"00:22.455 ","End":"00:24.315","Text":"when x is less than 1."},{"Start":"00:24.315 ","End":"00:29.475","Text":"Let\u0027s compute f composed with g. Remember that f composed with"},{"Start":"00:29.475 ","End":"00:35.250","Text":"g for some value x is equal to f of g of x."},{"Start":"00:35.250 ","End":"00:37.995","Text":"The first function we\u0027re going to apply is g."},{"Start":"00:37.995 ","End":"00:41.820","Text":"Let\u0027s divide according to cases in the piece wise."},{"Start":"00:41.820 ","End":"00:44.855","Text":"Let\u0027s take first of all x bigger or equal to 1,"},{"Start":"00:44.855 ","End":"00:47.345","Text":"and then we\u0027ll take x less than 1."},{"Start":"00:47.345 ","End":"00:50.690","Text":"First x bigger or equal to 1."},{"Start":"00:50.690 ","End":"00:56.720","Text":"In this case, what we get is that f of g of x is f of."},{"Start":"00:56.720 ","End":"00:59.000","Text":"Now x bigger or equal to 1,"},{"Start":"00:59.000 ","End":"01:00.245","Text":"g of x is 3."},{"Start":"01:00.245 ","End":"01:03.735","Text":"This is f of 3 and f of 3,"},{"Start":"01:03.735 ","End":"01:08.210","Text":"3 falls under the case where it\u0027s smaller than 5."},{"Start":"01:08.210 ","End":"01:13.235","Text":"We get 4 times 3 plus 3, which equal to 15."},{"Start":"01:13.235 ","End":"01:18.035","Text":"But if x is less than 1 and the same f of g of x,"},{"Start":"01:18.035 ","End":"01:20.225","Text":"g of x is now 2."},{"Start":"01:20.225 ","End":"01:24.265","Text":"What we get is f of 2 and f of 2,"},{"Start":"01:24.265 ","End":"01:26.595","Text":"2 is still less than 5."},{"Start":"01:26.595 ","End":"01:29.720","Text":"It\u0027s 4 times 2 plus 3,"},{"Start":"01:29.720 ","End":"01:32.750","Text":"which is 8 plus 3, which is 11."},{"Start":"01:32.750 ","End":"01:37.315","Text":"What we\u0027re going to do is write f compose g in piecewise form."},{"Start":"01:37.315 ","End":"01:42.065","Text":"F composed g of x is equal to,"},{"Start":"01:42.065 ","End":"01:46.565","Text":"in the case x bigger or equal to 1 is equal to 15 and"},{"Start":"01:46.565 ","End":"01:53.790","Text":"11 in the case where x is less than 1. That\u0027s it. We\u0027re done."}],"ID":4380}],"Thumbnail":null,"ID":1182},{"Name":"Even and Odd Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Even and Odd Functions","Duration":"7m 37s","ChapterTopicVideoID":13850,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.715","Text":"In this clip, I\u0027m going to explain about odd functions and even functions."},{"Start":"00:05.715 ","End":"00:06.890","Text":"Haven\u0027t defined it yet,"},{"Start":"00:06.890 ","End":"00:09.510","Text":"so the names may not mean anything to you,"},{"Start":"00:09.510 ","End":"00:12.540","Text":"but these will be 2 special functions."},{"Start":"00:12.540 ","End":"00:16.260","Text":"Anything that doesn\u0027t fall into 1 of these categories will"},{"Start":"00:16.260 ","End":"00:19.665","Text":"just be called a general function in this context."},{"Start":"00:19.665 ","End":"00:21.465","Text":"Let\u0027s take 1 of them."},{"Start":"00:21.465 ","End":"00:24.810","Text":"I\u0027d rather go with the even 1 first."},{"Start":"00:24.810 ","End":"00:29.580","Text":"I\u0027ll just give you an abstract definition of when a function f is called even."},{"Start":"00:29.580 ","End":"00:35.400","Text":"F is called even if the following condition holds."},{"Start":"00:35.400 ","End":"00:42.260","Text":"F of x equals f of minus x for all x,"},{"Start":"00:42.260 ","End":"00:45.485","Text":"which means that if I take any number"},{"Start":"00:45.485 ","End":"00:49.745","Text":"or it\u0027s negative and substitute them in the function,"},{"Start":"00:49.745 ","End":"00:51.635","Text":"I\u0027ll get the same thing."},{"Start":"00:51.635 ","End":"00:53.585","Text":"That\u0027s the abstract definition,"},{"Start":"00:53.585 ","End":"00:55.910","Text":"and now, we\u0027ll go with some examples."},{"Start":"00:55.910 ","End":"00:58.325","Text":"In fact, I\u0027ll write the 3 examples,"},{"Start":"00:58.325 ","End":"01:00.280","Text":"since these came from an exercise which said,"},{"Start":"01:00.280 ","End":"01:04.710","Text":"show that the following 3 functions are even functions."},{"Start":"01:04.990 ","End":"01:07.705","Text":"I\u0027ll just write them as 1,"},{"Start":"01:07.705 ","End":"01:09.315","Text":"2, and 3."},{"Start":"01:09.315 ","End":"01:15.645","Text":"The first one is f of x equals x squared."},{"Start":"01:15.645 ","End":"01:23.585","Text":"The second one will be f of x equals x^4. You know what?"},{"Start":"01:23.585 ","End":"01:26.750","Text":"To be correct, I should use different letters for the different functions."},{"Start":"01:26.750 ","End":"01:29.660","Text":"I\u0027ll change that f into a g, hang on."},{"Start":"01:29.660 ","End":"01:35.180","Text":"The third one, we\u0027ll have h of x is equal to 1 over"},{"Start":"01:35.180 ","End":"01:41.020","Text":"x squared plus the absolute value of x plus 2."},{"Start":"01:41.020 ","End":"01:45.630","Text":"Let\u0027s just test according to the definition that it holds,"},{"Start":"01:45.630 ","End":"01:47.420","Text":"then I can say that each of these, that f,"},{"Start":"01:47.420 ","End":"01:49.520","Text":"g, and h are all even functions."},{"Start":"01:49.520 ","End":"01:52.150","Text":"We\u0027ll take number 1 first."},{"Start":"01:52.150 ","End":"01:58.070","Text":"What we often do in inequality is start from one side and end up in the other side."},{"Start":"01:58.070 ","End":"02:02.535","Text":"I\u0027d like to start with the right-hand side of this part."},{"Start":"02:02.535 ","End":"02:04.650","Text":"To start with f of minus x,"},{"Start":"02:04.650 ","End":"02:07.100","Text":"and by algebraic manipulation,"},{"Start":"02:07.100 ","End":"02:08.510","Text":"end up with f of x."},{"Start":"02:08.510 ","End":"02:10.560","Text":"Here it goes, the first one,"},{"Start":"02:10.560 ","End":"02:13.140","Text":"f of minus x."},{"Start":"02:13.140 ","End":"02:19.105","Text":"We replace x by minus x equals minus x squared."},{"Start":"02:19.105 ","End":"02:22.100","Text":"Now, minus times minus is a plus,"},{"Start":"02:22.100 ","End":"02:24.335","Text":"so this is equal to x squared,"},{"Start":"02:24.335 ","End":"02:27.815","Text":"and this is just equal to f of x."},{"Start":"02:27.815 ","End":"02:29.900","Text":"If you look at the first and last,"},{"Start":"02:29.900 ","End":"02:34.325","Text":"we have that f of minus x equals f of x."},{"Start":"02:34.325 ","End":"02:39.065","Text":"Of course, here I didn\u0027t use any particular x, this works for any x."},{"Start":"02:39.065 ","End":"02:41.180","Text":"Number 1 is done,"},{"Start":"02:41.180 ","End":"02:43.875","Text":"that is even, f is even."},{"Start":"02:43.875 ","End":"02:45.420","Text":"Let\u0027s try number 2,"},{"Start":"02:45.420 ","End":"02:47.880","Text":"g of x is x^4."},{"Start":"02:47.880 ","End":"02:55.355","Text":"Let\u0027s see what happens to g of minus x. G of minus x is minus x^4."},{"Start":"02:55.355 ","End":"02:57.710","Text":"Again, minus to the power of 4,"},{"Start":"02:57.710 ","End":"02:58.820","Text":"minus, minus, minus,"},{"Start":"02:58.820 ","End":"03:02.675","Text":"minus is a plus, it\u0027s plus x^4."},{"Start":"03:02.675 ","End":"03:07.235","Text":"At this point you might see where someone got the idea even functions,"},{"Start":"03:07.235 ","End":"03:09.470","Text":"because all the even powers of x,"},{"Start":"03:09.470 ","End":"03:14.285","Text":"whether it\u0027s x squared or x^4 or x^18,"},{"Start":"03:14.285 ","End":"03:18.380","Text":"they\u0027ll all have this property that they\u0027ll be even functions."},{"Start":"03:18.380 ","End":"03:23.160","Text":"The even powers of x are the main examples of even functions."},{"Start":"03:23.160 ","End":"03:25.335","Text":"That accounted in fact for 1 and 2."},{"Start":"03:25.335 ","End":"03:26.540","Text":"I didn\u0027t finish this off,"},{"Start":"03:26.540 ","End":"03:28.760","Text":"and of course when you\u0027ve got x^4,"},{"Start":"03:28.760 ","End":"03:31.445","Text":"that equals g of x from here."},{"Start":"03:31.445 ","End":"03:38.030","Text":"Again, we have the property of f of minus x is the same thing as f of x,"},{"Start":"03:38.030 ","End":"03:41.690","Text":"only this time it was g. Let\u0027s go for h. Looks a bit longer,"},{"Start":"03:41.690 ","End":"03:43.835","Text":"but shouldn\u0027t be too difficult."},{"Start":"03:43.835 ","End":"03:46.375","Text":"H of minus x,"},{"Start":"03:46.375 ","End":"03:50.135","Text":"so we substitute minus x instead of x,"},{"Start":"03:50.135 ","End":"03:54.560","Text":"is 1 over minus x squared,"},{"Start":"03:54.560 ","End":"03:58.850","Text":"and then the absolute value of minus x,"},{"Start":"03:58.850 ","End":"04:00.940","Text":"and then plus 2."},{"Start":"04:00.940 ","End":"04:02.570","Text":"Now, what is this equal to?"},{"Start":"04:02.570 ","End":"04:04.700","Text":"Again, we have minus x squared,"},{"Start":"04:04.700 ","End":"04:05.780","Text":"which is x squared."},{"Start":"04:05.780 ","End":"04:09.260","Text":"This is equal to 1 over x squared."},{"Start":"04:09.260 ","End":"04:15.125","Text":"Now, the absolute value of minus x and the absolute value of x are the same."},{"Start":"04:15.125 ","End":"04:19.655","Text":"I mean, the absolute value of minus 7 and the absolute value of 7 are the same."},{"Start":"04:19.655 ","End":"04:20.870","Text":"If it was the other way around,"},{"Start":"04:20.870 ","End":"04:24.770","Text":"the absolute value of plus 8 is the absolute value of minus 8."},{"Start":"04:24.770 ","End":"04:26.150","Text":"Plus 8 and minus 8,"},{"Start":"04:26.150 ","End":"04:28.020","Text":"or plus 7 and minus 7,"},{"Start":"04:28.020 ","End":"04:30.170","Text":"or any number and its minus,"},{"Start":"04:30.170 ","End":"04:32.060","Text":"they have the same absolute value."},{"Start":"04:32.060 ","End":"04:36.565","Text":"I could just write this simply as absolute value of x,"},{"Start":"04:36.565 ","End":"04:37.870","Text":"and 2 is just 2,"},{"Start":"04:37.870 ","End":"04:39.520","Text":"it doesn\u0027t depend on x."},{"Start":"04:39.520 ","End":"04:40.895","Text":"This is what we get,"},{"Start":"04:40.895 ","End":"04:44.585","Text":"and this is precisely the definition of h of x."},{"Start":"04:44.585 ","End":"04:49.910","Text":"Once again, we started with a general x and showed that h of minus x is h of x,"},{"Start":"04:49.910 ","End":"04:52.655","Text":"so that makes h even also."},{"Start":"04:52.655 ","End":"04:56.020","Text":"All these 3 things are even functions."},{"Start":"04:56.020 ","End":"04:59.900","Text":"Now, let\u0027s go on to the odd functions."},{"Start":"04:59.900 ","End":"05:04.110","Text":"F is called odd if f of,"},{"Start":"05:04.110 ","End":"05:08.180","Text":"again, I try to put minus x instead of x,"},{"Start":"05:08.180 ","End":"05:12.230","Text":"where as previously it came out the same as f of x,"},{"Start":"05:12.230 ","End":"05:14.635","Text":"this time it\u0027s going to come out the negative,"},{"Start":"05:14.635 ","End":"05:20.805","Text":"is negative f of x. I\u0027ll give 3 examples here."},{"Start":"05:20.805 ","End":"05:22.860","Text":"The 3 examples will be;"},{"Start":"05:22.860 ","End":"05:25.380","Text":"the first 1 is f of x is x cubed,"},{"Start":"05:25.380 ","End":"05:29.475","Text":"second one, g of x will be x^5."},{"Start":"05:29.475 ","End":"05:32.385","Text":"Notice that 3 and 5 are odd numbers,"},{"Start":"05:32.385 ","End":"05:39.555","Text":"and h of x which will be 1 over x cubed,"},{"Start":"05:39.555 ","End":"05:43.200","Text":"that\u0027s a 3 there, plus x."},{"Start":"05:43.200 ","End":"05:50.435","Text":"The claim is or the request is to show that each of these 3 is an odd function."},{"Start":"05:50.435 ","End":"05:53.125","Text":"Let\u0027s try it out on the first one."},{"Start":"05:53.125 ","End":"05:56.715","Text":"Number 1, I start with f of minus x."},{"Start":"05:56.715 ","End":"05:58.735","Text":"When f of x is x cubed,"},{"Start":"05:58.735 ","End":"06:03.800","Text":"f of minus x is minus x cubed."},{"Start":"06:03.800 ","End":"06:07.340","Text":"Now, 3 is an odd number, minus, minus,"},{"Start":"06:07.340 ","End":"06:10.485","Text":"minus is going to give me a minus,"},{"Start":"06:10.485 ","End":"06:13.455","Text":"or if you like it\u0027s minus 1 cubed,"},{"Start":"06:13.455 ","End":"06:17.360","Text":"and minus 1 times minus 1 times minus 1 is minus 1."},{"Start":"06:17.360 ","End":"06:20.674","Text":"Basically an odd number of minuses will give us a minus."},{"Start":"06:20.674 ","End":"06:22.975","Text":"That\u0027s the thing about the odd."},{"Start":"06:22.975 ","End":"06:27.410","Text":"The name derives from the fact that the odd powers of x are odd functions."},{"Start":"06:27.410 ","End":"06:33.590","Text":"This gives us minus cubed is still minus, and x cubed."},{"Start":"06:33.590 ","End":"06:35.240","Text":"These 2 things are totally different."},{"Start":"06:35.240 ","End":"06:38.360","Text":"Here is minus x all raised to the power of 3,"},{"Start":"06:38.360 ","End":"06:41.975","Text":"and this is just x^3 or x cubed and then a minus,"},{"Start":"06:41.975 ","End":"06:44.795","Text":"but it comes out the same because it\u0027s an odd function."},{"Start":"06:44.795 ","End":"06:46.520","Text":"This is equal,"},{"Start":"06:46.520 ","End":"06:48.110","Text":"if I leave the minus here,"},{"Start":"06:48.110 ","End":"06:51.165","Text":"what I\u0027m left with is f of x."},{"Start":"06:51.165 ","End":"06:53.239","Text":"If you now look at this formula,"},{"Start":"06:53.239 ","End":"06:57.290","Text":"f of minus x is minus f of x from here to here,"},{"Start":"06:57.290 ","End":"06:58.775","Text":"and through a general x,"},{"Start":"06:58.775 ","End":"07:00.710","Text":"so yes, this is odd."},{"Start":"07:00.710 ","End":"07:03.650","Text":"In the second one, we had x^5."},{"Start":"07:03.650 ","End":"07:09.630","Text":"If we take g of minus x, it\u0027s minus x^5."},{"Start":"07:09.630 ","End":"07:11.810","Text":"Once again, because 5 is an odd number,"},{"Start":"07:11.810 ","End":"07:13.250","Text":"minus, minus, minus, minus,"},{"Start":"07:13.250 ","End":"07:17.655","Text":"minus is just minus x^5."},{"Start":"07:17.655 ","End":"07:21.825","Text":"Since just the x^5 part is g of x,"},{"Start":"07:21.825 ","End":"07:25.230","Text":"what we have here is minus g of x. G fits"},{"Start":"07:25.230 ","End":"07:29.175","Text":"the same formula as f for the definition about functions,"},{"Start":"07:29.175 ","End":"07:31.530","Text":"so g is odd also."},{"Start":"07:31.530 ","End":"07:33.150","Text":"Perhaps I\u0027ll just write this at the side,"},{"Start":"07:33.150 ","End":"07:38.080","Text":"this means that f is odd, g is odd."}],"ID":14649},{"Watched":false,"Name":"Exercise 1","Duration":"1m 15s","ChapterTopicVideoID":4372,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we\u0027re given the graphs of"},{"Start":"00:02.550 ","End":"00:06.780","Text":"several functions and we have to say which of these functions is 1 to 1."},{"Start":"00:06.780 ","End":"00:10.230","Text":"Now remember, a function is 1 to 1 if"},{"Start":"00:10.230 ","End":"00:14.775","Text":"a horizontal line can cross the function at most 1 time."},{"Start":"00:14.775 ","End":"00:16.920","Text":"Let\u0027s look at both a and b."},{"Start":"00:16.920 ","End":"00:22.800","Text":"In a, we see that any horizontal line will only cut the function at most 1 time."},{"Start":"00:22.800 ","End":"00:24.090","Text":"It can\u0027t cut twice,"},{"Start":"00:24.090 ","End":"00:25.350","Text":"but in b,"},{"Start":"00:25.350 ","End":"00:28.830","Text":"a horizontal line can cut the function twice."},{"Start":"00:28.830 ","End":"00:31.305","Text":"For example, here and here."},{"Start":"00:31.305 ","End":"00:33.435","Text":"For example, if this was, say,"},{"Start":"00:33.435 ","End":"00:38.425","Text":"minus 2 and this was 2 and this value was 1,"},{"Start":"00:38.425 ","End":"00:44.110","Text":"then we would have that f of minus 2 equals f of 2,"},{"Start":"00:44.110 ","End":"00:46.950","Text":"but minus 2 is not equal 2."},{"Start":"00:46.950 ","End":"00:50.450","Text":"It\u0027s not 1 to 1 on the algebraic definition also."},{"Start":"00:50.450 ","End":"00:53.545","Text":"In other words, so far we have that a,"},{"Start":"00:53.545 ","End":"00:55.500","Text":"we can say, yes,"},{"Start":"00:55.500 ","End":"00:57.285","Text":"it is 1 to 1."},{"Start":"00:57.285 ","End":"00:59.490","Text":"In b, the answer is no."},{"Start":"00:59.490 ","End":"01:01.900","Text":"Continuing to c and d,"},{"Start":"01:01.900 ","End":"01:04.910","Text":"it\u0027s clear that neither of these is 1 to 1 because,"},{"Start":"01:04.910 ","End":"01:09.490","Text":"for example, a horizontal line here will cross twice."},{"Start":"01:09.490 ","End":"01:12.650","Text":"Similarly here, not only can it cross twice,"},{"Start":"01:12.650 ","End":"01:15.980","Text":"a horizontal line will cut many times here,"},{"Start":"01:15.980 ","End":"01:18.415","Text":"here, here, here, and so on."},{"Start":"01:18.415 ","End":"01:20.705","Text":"For these we can answer."},{"Start":"01:20.705 ","End":"01:23.420","Text":"This is no, not 1 to 1,"},{"Start":"01:23.420 ","End":"01:25.240","Text":"and also here it\u0027s no."},{"Start":"01:25.240 ","End":"01:26.760","Text":"In example e,"},{"Start":"01:26.760 ","End":"01:33.815","Text":"we also see that it is 1 to 1 because any horizontal line will cut at most once."},{"Start":"01:33.815 ","End":"01:37.070","Text":"Of course, a horizontal line might not cut at all."},{"Start":"01:37.070 ","End":"01:40.730","Text":"But the point is that it must not cut more than once."},{"Start":"01:40.730 ","End":"01:45.505","Text":"E is definitely 1 to 1 so is f,"},{"Start":"01:45.505 ","End":"01:50.300","Text":"any horizontal line that we control will not cut the graph twice."},{"Start":"01:50.300 ","End":"01:55.800","Text":"F is also 1 to 1 and we\u0027re done for this exercise."}],"ID":4381},{"Watched":false,"Name":"Exercise 2","Duration":"1m 33s","ChapterTopicVideoID":4373,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.360 ","End":"00:06.840","Text":"odd or neither, and to give reasons for our answer,"},{"Start":"00:06.840 ","End":"00:09.465","Text":"and the function is as described here."},{"Start":"00:09.465 ","End":"00:19.485","Text":"Recall that an even function means that f of minus x equals f of x for all x,"},{"Start":"00:19.485 ","End":"00:28.835","Text":"and odd means that f of minus x equals minus f of x for all x."},{"Start":"00:28.835 ","End":"00:32.930","Text":"In either case, we substitute minus x for x in the function,"},{"Start":"00:32.930 ","End":"00:34.405","Text":"so in this case,"},{"Start":"00:34.405 ","End":"00:38.020","Text":"f of minus x equals,"},{"Start":"00:38.020 ","End":"00:41.375","Text":"we just substitute minus x instead of x in the function,"},{"Start":"00:41.375 ","End":"00:48.340","Text":"so we get minus x to the 4th plus minus x to the 10th."},{"Start":"00:48.340 ","End":"00:52.450","Text":"Now, minus x is simply minus 1 times x,"},{"Start":"00:52.450 ","End":"00:55.705","Text":"so we get minus 1 to the power of 4,"},{"Start":"00:55.705 ","End":"01:00.935","Text":"x to the power of 4 plus minus 1 to the power of 10,"},{"Start":"01:00.935 ","End":"01:02.855","Text":"x to the power of 10."},{"Start":"01:02.855 ","End":"01:04.970","Text":"Now, because 4 is an even number,"},{"Start":"01:04.970 ","End":"01:07.525","Text":"minus 1 to the 4th is just 1,"},{"Start":"01:07.525 ","End":"01:10.140","Text":"leaving us here with just x to the 4th."},{"Start":"01:10.140 ","End":"01:14.265","Text":"Similarly, the second term is just x to the 10th,"},{"Start":"01:14.265 ","End":"01:15.584","Text":"and if you notice,"},{"Start":"01:15.584 ","End":"01:19.655","Text":"this thing is exactly equal to the definition of the original f of x,"},{"Start":"01:19.655 ","End":"01:22.710","Text":"so this equals f of x."},{"Start":"01:22.710 ","End":"01:28.200","Text":"In other words, f of minus x is equal to f of x,"},{"Start":"01:28.200 ","End":"01:33.750","Text":"and we see from above that this makes the function even, and we\u0027re done."}],"ID":4382},{"Watched":false,"Name":"Exercise 3","Duration":"56s","ChapterTopicVideoID":4374,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.465","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.465 ","End":"00:04.500","Text":"odd or neither,"},{"Start":"00:04.500 ","End":"00:06.570","Text":"and to give reasons for our answer."},{"Start":"00:06.570 ","End":"00:11.640","Text":"Recall that an odd function is 1 for which f of"},{"Start":"00:11.640 ","End":"00:19.380","Text":"minus x is equal to minus f(x) for all x and that\u0027s an even function,"},{"Start":"00:19.380 ","End":"00:25.800","Text":"is 1 for which f of minus x equals f(x) for all x."},{"Start":"00:25.800 ","End":"00:34.635","Text":"In our case, our function y equals 1 simply means that f(x) is equal to 1 for all x,"},{"Start":"00:34.635 ","End":"00:40.085","Text":"f(-x) is also equal to 1 because it doesn\u0027t matter what we put for x."},{"Start":"00:40.085 ","End":"00:41.540","Text":"This is a constant function."},{"Start":"00:41.540 ","End":"00:43.280","Text":"f of minus x is 1,"},{"Start":"00:43.280 ","End":"00:45.545","Text":"which is the same as f(x)."},{"Start":"00:45.545 ","End":"00:47.300","Text":"Since this is the case,"},{"Start":"00:47.300 ","End":"00:51.050","Text":"we see that f of minus x is f(x),"},{"Start":"00:51.050 ","End":"00:53.480","Text":"which is exactly the definition of even."},{"Start":"00:53.480 ","End":"00:56.880","Text":"This is an even function and we\u0027re done."}],"ID":4383},{"Watched":false,"Name":"Exercise 4","Duration":"1m 43s","ChapterTopicVideoID":4375,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.450 ","End":"00:07.910","Text":"odd or neither and to give reasons for our answer. What is the function?"},{"Start":"00:07.910 ","End":"00:12.480","Text":"It\u0027s written in the form of y equals something and it would\u0027ve been better to write it as"},{"Start":"00:12.480 ","End":"00:17.280","Text":"f of x using the function notation equals the same thing as there,"},{"Start":"00:17.280 ","End":"00:21.455","Text":"1 over x plus the cube root of x."},{"Start":"00:21.455 ","End":"00:26.450","Text":"Recall that an odd function is 1 for which f of"},{"Start":"00:26.450 ","End":"00:31.670","Text":"minus x equals minus f of x for all x,"},{"Start":"00:31.670 ","End":"00:39.710","Text":"and that an even function is 1 for which f of minus x is the same as f of x for all x."},{"Start":"00:39.710 ","End":"00:41.270","Text":"Let\u0027s see what happens in our case."},{"Start":"00:41.270 ","End":"00:44.785","Text":"What we have to do is to substitute minus x for x,"},{"Start":"00:44.785 ","End":"00:51.245","Text":"so f of minus x is equal to 1 over minus x,"},{"Start":"00:51.245 ","End":"00:57.380","Text":"which is putting minus x here plus the cube root of minus x."},{"Start":"00:57.380 ","End":"01:03.305","Text":"The first term is simply minus 1 over x and the second term is"},{"Start":"01:03.305 ","End":"01:09.275","Text":"minus the cube root of x and if you\u0027re wondering why this is so?"},{"Start":"01:09.275 ","End":"01:15.830","Text":"Perhaps look at a simple example where the cube root of minus 8 is minus 2,"},{"Start":"01:15.830 ","End":"01:19.700","Text":"which is minus the cube root of 8."},{"Start":"01:19.700 ","End":"01:28.540","Text":"Take minus outside the brackets and we get minus 1 over x plus the cube root of x,"},{"Start":"01:28.540 ","End":"01:33.195","Text":"and that\u0027s simply equals minus f of x."},{"Start":"01:33.195 ","End":"01:39.275","Text":"What we have is that f of minus x equals minus f of x,"},{"Start":"01:39.275 ","End":"01:43.930","Text":"which makes this function odd and we\u0027re done."}],"ID":4384},{"Watched":false,"Name":"Exercise 5","Duration":"2m 38s","ChapterTopicVideoID":4376,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.285 ","End":"00:06.660","Text":"odd or neither, and to give reasons for our answer."},{"Start":"00:06.660 ","End":"00:09.795","Text":"The function is given in the form of y equals,"},{"Start":"00:09.795 ","End":"00:13.800","Text":"and it\u0027s really better to write it in functional notation, in other words,"},{"Start":"00:13.800 ","End":"00:18.300","Text":"f of x equals x squared plus x cubed."},{"Start":"00:18.300 ","End":"00:20.475","Text":"Now a reminder of odd and even."},{"Start":"00:20.475 ","End":"00:28.500","Text":"An even function means that f of minus x is equal to f of x for all x,"},{"Start":"00:28.500 ","End":"00:37.650","Text":"and an odd function means that f of minus x equals minus f of x for all x."},{"Start":"00:37.650 ","End":"00:42.085","Text":"So let\u0027s see what happens in our case with f of minus x."},{"Start":"00:42.085 ","End":"00:44.960","Text":"To get this, we simply substitute minus x instead"},{"Start":"00:44.960 ","End":"00:47.690","Text":"of x in the original function where x is."},{"Start":"00:47.690 ","End":"00:54.710","Text":"We get minus x squared plus minus x cubed,"},{"Start":"00:54.710 ","End":"00:57.170","Text":"minus x is simply minus 1 times x,"},{"Start":"00:57.170 ","End":"01:06.350","Text":"so this gives us minus 1 squared times x squared plus minus 1 cubed times x cubed."},{"Start":"01:06.350 ","End":"01:08.000","Text":"Since 2 is even,"},{"Start":"01:08.000 ","End":"01:09.350","Text":"minus 1 squared is 1,"},{"Start":"01:09.350 ","End":"01:12.740","Text":"so this is just x squared and 3 is odd,"},{"Start":"01:12.740 ","End":"01:16.020","Text":"so minus 1 cubed is minus 1,"},{"Start":"01:16.020 ","End":"01:19.115","Text":"so we get x squared minus x cubed."},{"Start":"01:19.115 ","End":"01:24.980","Text":"Now, notice that this is not the same as f of x,"},{"Start":"01:24.980 ","End":"01:27.890","Text":"but it\u0027s not the same as minus f of x either."},{"Start":"01:27.890 ","End":"01:29.485","Text":"It\u0027s not simply the negation."},{"Start":"01:29.485 ","End":"01:31.880","Text":"It looks like in our case,"},{"Start":"01:31.880 ","End":"01:35.510","Text":"it\u0027s neither odd nor even."},{"Start":"01:35.510 ","End":"01:38.990","Text":"However, this is insufficient for the case of neither,"},{"Start":"01:38.990 ","End":"01:42.035","Text":"we are expected to provide a counterexample,"},{"Start":"01:42.035 ","End":"01:47.870","Text":"meaning a value of x for which f of minus x is not equal to f of x or minus f of x."},{"Start":"01:47.870 ","End":"01:51.620","Text":"Let\u0027s just try and substitute x equals 1."},{"Start":"01:51.620 ","End":"01:59.945","Text":"Now, f of minus x is equal to minus 1 squared plus minus 1 cubed,"},{"Start":"01:59.945 ","End":"02:04.880","Text":"and this equals 1 minus 1, which equals 0."},{"Start":"02:04.880 ","End":"02:06.260","Text":"On the other hand,"},{"Start":"02:06.260 ","End":"02:13.120","Text":"f of x is equal to 1 squared plus 1 cubed equals 2,"},{"Start":"02:13.120 ","End":"02:19.385","Text":"and so we see that f of minus x is not equal to f of x,"},{"Start":"02:19.385 ","End":"02:24.320","Text":"just because 0 is not equal to 2."},{"Start":"02:24.320 ","End":"02:34.370","Text":"But also f of minus x is not equal to minus f of x because 0 is not equal to minus 2,"},{"Start":"02:34.370 ","End":"02:38.940","Text":"and this gives us our justification and we\u0027re done."}],"ID":4385},{"Watched":false,"Name":"Exercise 6","Duration":"1m 59s","ChapterTopicVideoID":4377,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.210 ","End":"00:05.670","Text":"odd or neither and to give reasons."},{"Start":"00:05.670 ","End":"00:11.985","Text":"The function is y equals x cubed plus 4x over the absolute value of x plus 1."},{"Start":"00:11.985 ","End":"00:14.280","Text":"This is written in the y notation."},{"Start":"00:14.280 ","End":"00:21.045","Text":"We can just write this as equal to f of x. I want to remind you what odd and even means."},{"Start":"00:21.045 ","End":"00:29.370","Text":"A function f is called odd if f of minus x equals minus f of x for all x."},{"Start":"00:29.370 ","End":"00:34.995","Text":"It\u0027s called even if f of minus x is equal to f of x."},{"Start":"00:34.995 ","End":"00:36.360","Text":"Again, for all x."},{"Start":"00:36.360 ","End":"00:39.505","Text":"In both cases, we examine f of minus x."},{"Start":"00:39.505 ","End":"00:44.320","Text":"In our case, we see that f of minus x is equal to,"},{"Start":"00:44.320 ","End":"00:49.685","Text":"I Just substitute minus x for x in the original f and I get"},{"Start":"00:49.685 ","End":"01:00.275","Text":"minus x cubed plus 4 times minus x over the absolute value of minus x plus 1."},{"Start":"01:00.275 ","End":"01:01.940","Text":"Let\u0027s simplify this a bit,"},{"Start":"01:01.940 ","End":"01:04.985","Text":"minus x cubed and 3 is an odd number,"},{"Start":"01:04.985 ","End":"01:11.440","Text":"is just minus of x cubed and 4 times minus x is minus 4x,"},{"Start":"01:11.440 ","End":"01:16.655","Text":"absolute value of minus x is the same as the absolute value of x."},{"Start":"01:16.655 ","End":"01:20.590","Text":"For example, absolute value of minus 4 and absolute value of 4 are the same."},{"Start":"01:20.590 ","End":"01:24.935","Text":"So this is just absolute value of x plus 1."},{"Start":"01:24.935 ","End":"01:27.920","Text":"Now, we can take minus 1 as"},{"Start":"01:27.920 ","End":"01:32.315","Text":"a common factor out of the numerator and we\u0027ll put it at the side of the fraction."},{"Start":"01:32.315 ","End":"01:40.745","Text":"We get minus x cubed plus 4x over absolute value of x plus 1."},{"Start":"01:40.745 ","End":"01:43.730","Text":"If we look at this expression to the right of the minus,"},{"Start":"01:43.730 ","End":"01:46.595","Text":"it\u0027s exactly the same as the original f of x."},{"Start":"01:46.595 ","End":"01:50.180","Text":"Here we get minus f of x."},{"Start":"01:50.180 ","End":"01:55.130","Text":"In other words, we started out with f of minus x and ended up with minus f of x."},{"Start":"01:55.130 ","End":"02:00.150","Text":"By the definition, this makes this function odd. We\u0027re done."}],"ID":4386},{"Watched":false,"Name":"Exercise 7","Duration":"1m 38s","ChapterTopicVideoID":4379,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.060 ","End":"00:05.730","Text":"odd, or neither, and to give reasons."},{"Start":"00:05.730 ","End":"00:10.860","Text":"In our case, y equals 1/2 of e to the x plus e to the minus x."},{"Start":"00:10.860 ","End":"00:12.615","Text":"Using the y notation,"},{"Start":"00:12.615 ","End":"00:15.480","Text":"and I prefer the f of x notation."},{"Start":"00:15.480 ","End":"00:20.010","Text":"I\u0027ll call this f of x. Let\u0027s remember when a function f is odd or even,"},{"Start":"00:20.010 ","End":"00:27.540","Text":"f is considered to be an odd function if f of minus x equals minus f of x."},{"Start":"00:27.540 ","End":"00:34.245","Text":"It\u0027s considered to be even if f of minus x equals f of x."},{"Start":"00:34.245 ","End":"00:36.950","Text":"We have to look at f of minus x in either case."},{"Start":"00:36.950 ","End":"00:41.540","Text":"In our case, we get f of minus x equals,"},{"Start":"00:41.540 ","End":"00:46.970","Text":"substitute the set of x minus x in the original function and we get 1/2 of"},{"Start":"00:46.970 ","End":"00:53.170","Text":"e to the minus x plus e to the minus minus x."},{"Start":"00:53.170 ","End":"00:58.400","Text":"This equals 1/2 of e to the minus x plus,"},{"Start":"00:58.400 ","End":"01:01.250","Text":"now minus minus x is the same as x."},{"Start":"01:01.250 ","End":"01:03.710","Text":"This very much looks like the original function."},{"Start":"01:03.710 ","End":"01:06.430","Text":"In fact, if we just changed the order of these 2 terms,"},{"Start":"01:06.430 ","End":"01:12.140","Text":"we get 1/2 of e to the x plus e to the minus x,"},{"Start":"01:12.140 ","End":"01:16.565","Text":"which is exactly the same as the original f of x."},{"Start":"01:16.565 ","End":"01:21.380","Text":"In other words, f of minus x is equal to f of x."},{"Start":"01:21.380 ","End":"01:24.725","Text":"It fits the definition of even."},{"Start":"01:24.725 ","End":"01:28.520","Text":"We\u0027re done, but I would like to mention that this particular function has"},{"Start":"01:28.520 ","End":"01:33.080","Text":"a name and it\u0027s called the hyperbolic cosine of x,"},{"Start":"01:33.080 ","End":"01:38.730","Text":"written as cosine h of x. That\u0027s all."}],"ID":4388},{"Watched":false,"Name":"Exercise 8","Duration":"1m 54s","ChapterTopicVideoID":4380,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.210 ","End":"00:05.790","Text":"odd or neither and to give reasons."},{"Start":"00:05.790 ","End":"00:11.820","Text":"The function here is y equals 1/2 of e to the x minus e to the minus x."},{"Start":"00:11.820 ","End":"00:16.470","Text":"Let\u0027s use the functional notation f of x. I want to"},{"Start":"00:16.470 ","End":"00:22.245","Text":"remind you that a function f is called odd if f of minus x"},{"Start":"00:22.245 ","End":"00:27.390","Text":"is equal to minus f of x for all x and it\u0027s called"},{"Start":"00:27.390 ","End":"00:34.390","Text":"even if f of minus x is equal to f of x for all x."},{"Start":"00:34.390 ","End":"00:39.260","Text":"In our case, f of minus x is equal to,"},{"Start":"00:39.260 ","End":"00:42.710","Text":"just substitute minus x for x in the original function,"},{"Start":"00:42.710 ","End":"00:52.360","Text":"and we get 1/2 of e to the minus x minus e to the power of minus minus x."},{"Start":"00:52.360 ","End":"00:54.500","Text":"Let\u0027s do some simplification."},{"Start":"00:54.500 ","End":"01:02.900","Text":"This is equal to 1/2 of e to the minus x minus e to the power of x."},{"Start":"01:02.900 ","End":"01:05.780","Text":"This looks very similar to the original f of x,"},{"Start":"01:05.780 ","End":"01:08.599","Text":"but not quite because the order of the terms is reversed."},{"Start":"01:08.599 ","End":"01:11.765","Text":"But if we take minus outside the brackets,"},{"Start":"01:11.765 ","End":"01:13.445","Text":"we get minus 1/2,"},{"Start":"01:13.445 ","End":"01:15.680","Text":"and instead of a minus b,"},{"Start":"01:15.680 ","End":"01:17.540","Text":"we can write b minus a."},{"Start":"01:17.540 ","End":"01:19.895","Text":"In other words, e to the x,"},{"Start":"01:19.895 ","End":"01:22.850","Text":"minus e to the minus x."},{"Start":"01:22.850 ","End":"01:24.200","Text":"Now, after the minus,"},{"Start":"01:24.200 ","End":"01:26.410","Text":"we have exactly the original f of x,"},{"Start":"01:26.410 ","End":"01:29.360","Text":"so this is minus f of x."},{"Start":"01:29.360 ","End":"01:34.265","Text":"In other words, we have that f of minus x equals minus f of x,"},{"Start":"01:34.265 ","End":"01:38.100","Text":"which fits the definition of an odd function and we\u0027re"},{"Start":"01:38.100 ","End":"01:42.664","Text":"done except that I\u0027d like to mention that this particular function,"},{"Start":"01:42.664 ","End":"01:46.490","Text":"1/2 e to the x minus e to the minus x has a name,"},{"Start":"01:46.490 ","End":"01:50.160","Text":"and it\u0027s called the hyperbolic sine, written like this,"},{"Start":"01:50.160 ","End":"01:55.090","Text":"sine h of x. That\u0027s it."}],"ID":4389},{"Watched":false,"Name":"Exercise 9","Duration":"1m 9s","ChapterTopicVideoID":4381,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.885 ","End":"00:06.950","Text":"odd or neither and to give reasons for our answer."},{"Start":"00:06.950 ","End":"00:11.100","Text":"Y equals natural log of x plus x plus 1,"},{"Start":"00:11.100 ","End":"00:13.665","Text":"which we\u0027ll call f of x."},{"Start":"00:13.665 ","End":"00:15.960","Text":"Now remember, an odd function,"},{"Start":"00:15.960 ","End":"00:21.690","Text":"f is 1 for which f of minus x equals minus f of x"},{"Start":"00:21.690 ","End":"00:28.530","Text":"and it\u0027s called even if f of minus x equals f of x for all x."},{"Start":"00:28.530 ","End":"00:33.060","Text":"However, we should relate to the domain of the function and in our case,"},{"Start":"00:33.060 ","End":"00:37.615","Text":"the domain of the function is limited by natural log of x."},{"Start":"00:37.615 ","End":"00:40.700","Text":"Natural log is only defined for positive x."},{"Start":"00:40.700 ","End":"00:42.830","Text":"In other words x has to be bigger than 0."},{"Start":"00:42.830 ","End":"00:45.005","Text":"The x plus 1 is defined everywhere."},{"Start":"00:45.005 ","End":"00:50.269","Text":"Now if x is bigger than 0 and I\u0027m going to substitute f of minus x,"},{"Start":"00:50.269 ","End":"00:53.750","Text":"minus x will be negative and so I can\u0027t define"},{"Start":"00:53.750 ","End":"00:58.040","Text":"f because the natural log of a negative number is not defined."},{"Start":"00:58.040 ","End":"01:05.015","Text":"Basically I\u0027m stuck with computing f of minus x so the function can\u0027t be odd or even."},{"Start":"01:05.015 ","End":"01:09.820","Text":"I can immediately say that it\u0027s neither and we\u0027re done."}],"ID":4390},{"Watched":false,"Name":"Exercise 10","Duration":"1m 17s","ChapterTopicVideoID":4382,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:02.850 ","End":"00:05.115","Text":"odd, or neither, and to give reasons."},{"Start":"00:05.115 ","End":"00:10.755","Text":"In our case the function being y equals natural log squared of x plus x squared."},{"Start":"00:10.755 ","End":"00:14.130","Text":"This is in the y notation and I\u0027ll just write it as f of x."},{"Start":"00:14.130 ","End":"00:18.120","Text":"I want to remind you that a function f is called odd,"},{"Start":"00:18.120 ","End":"00:24.820","Text":"if f of minus x equals minus f of x for all x,"},{"Start":"00:24.820 ","End":"00:32.130","Text":"and it\u0027s known as even if f of minus x equals f of x for all x."},{"Start":"00:32.130 ","End":"00:35.070","Text":"We have to pay attention to the domain of the function."},{"Start":"00:35.070 ","End":"00:41.934","Text":"This natural log squared of x just simply means natural log of x all squared."},{"Start":"00:41.934 ","End":"00:45.100","Text":"But in any case, natural log of x has to be defined,"},{"Start":"00:45.100 ","End":"00:47.105","Text":"and that\u0027s really the only condition,"},{"Start":"00:47.105 ","End":"00:52.565","Text":"which means that x has to be positive for the natural log to be defined."},{"Start":"00:52.565 ","End":"00:54.725","Text":"Now if x has to be positive,"},{"Start":"00:54.725 ","End":"00:57.770","Text":"and we have to try and compute f of minus x,"},{"Start":"00:57.770 ","End":"01:00.890","Text":"this won\u0027t do because if x is positive,"},{"Start":"01:00.890 ","End":"01:02.570","Text":"minus x is negative,"},{"Start":"01:02.570 ","End":"01:04.635","Text":"this would be outside the domain,"},{"Start":"01:04.635 ","End":"01:09.650","Text":"so we can\u0027t even compute minus x because we\u0027d be computing the log of a negative number."},{"Start":"01:09.650 ","End":"01:13.040","Text":"Since we can\u0027t even compute f of minus x we have to say that"},{"Start":"01:13.040 ","End":"01:17.820","Text":"the function is neither odd nor even. We\u0027re done."}],"ID":4391},{"Watched":false,"Name":"Exercise 11","Duration":"1m 19s","ChapterTopicVideoID":4383,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.230","Text":"In this exercise,"},{"Start":"00:01.230 ","End":"00:03.585","Text":"we have to say whether the function is even,"},{"Start":"00:03.585 ","End":"00:04.875","Text":"odd, or neither,"},{"Start":"00:04.875 ","End":"00:06.495","Text":"and to give reasons."},{"Start":"00:06.495 ","End":"00:11.535","Text":"In this case, the function is f of x is sine x times cosine x."},{"Start":"00:11.535 ","End":"00:14.490","Text":"Now let\u0027s remember what odd and even mean."},{"Start":"00:14.490 ","End":"00:23.445","Text":"A function is called even if f of minus x is equal to f of x for all x."},{"Start":"00:23.445 ","End":"00:30.585","Text":"A function is called odd if f of minus x equals minus f of x,"},{"Start":"00:30.585 ","End":"00:32.145","Text":"again for all x."},{"Start":"00:32.145 ","End":"00:36.060","Text":"In both cases, we examine f of minus x and in our case,"},{"Start":"00:36.060 ","End":"00:39.810","Text":"we get that f of minus x equals,"},{"Start":"00:39.810 ","End":"00:42.720","Text":"substituting minus x for x here and here,"},{"Start":"00:42.720 ","End":"00:46.430","Text":"we get sine of minus x,"},{"Start":"00:46.430 ","End":"00:49.760","Text":"cosine of minus x."},{"Start":"00:49.760 ","End":"00:55.850","Text":"Now remember from trigonometry that sine of minus x is minus sine of x,"},{"Start":"00:55.850 ","End":"00:59.605","Text":"and that cosine of minus x is the same as cosine x,"},{"Start":"00:59.605 ","End":"01:02.955","Text":"so we get minus sine x cosine x."},{"Start":"01:02.955 ","End":"01:06.660","Text":"If you look at it, it\u0027s exactly the same as the definition of f of x,"},{"Start":"01:06.660 ","End":"01:10.715","Text":"so in the end we just get minus f of x."},{"Start":"01:10.715 ","End":"01:15.260","Text":"Since f of minus x equals minus f of x,"},{"Start":"01:15.260 ","End":"01:19.770","Text":"this makes our function odd and we\u0027re done."}],"ID":4392},{"Watched":false,"Name":"Exercise 12","Duration":"2m 20s","ChapterTopicVideoID":4385,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:03.645 ","End":"00:06.735","Text":"odd or neither and to give reasons."},{"Start":"00:06.735 ","End":"00:11.400","Text":"In our case, the function is written in the form of y and we could write this as"},{"Start":"00:11.400 ","End":"00:16.035","Text":"f of x. I\u0027d like to remind you what odd and even are."},{"Start":"00:16.035 ","End":"00:25.955","Text":"A function is called odd if f of minus x is equal to minus f of x,"},{"Start":"00:25.955 ","End":"00:34.880","Text":"and it\u0027s called even if f of minus x is equal to f of x."},{"Start":"00:34.880 ","End":"00:37.940","Text":"In both cases, we examined f of minus x."},{"Start":"00:37.940 ","End":"00:44.050","Text":"In our case, we get f of minus x is equal to,"},{"Start":"00:44.050 ","End":"00:47.215","Text":"and we substitute instead of x minus x,"},{"Start":"00:47.215 ","End":"00:54.065","Text":"it\u0027s equal to tangent of minus x over"},{"Start":"00:54.065 ","End":"01:02.555","Text":"minus x squared plus cosine of minus x."},{"Start":"01:02.555 ","End":"01:08.210","Text":"Now, I\u0027d like to remind you of a trigonometric formula that tangent of minus x"},{"Start":"01:08.210 ","End":"01:14.760","Text":"is the same as minus tangent of x. I\u0027ll demonstrate why for those who need to know."},{"Start":"01:15.160 ","End":"01:17.795","Text":"Now if you look at it closely,"},{"Start":"01:17.795 ","End":"01:23.210","Text":"this is simply equal to minus f of x because except for the minus,"},{"Start":"01:23.210 ","End":"01:26.165","Text":"this thing is the same as this thing here."},{"Start":"01:26.165 ","End":"01:32.000","Text":"What we conclude is that f of minus x is minus f of x,"},{"Start":"01:32.000 ","End":"01:35.960","Text":"and this exactly fits the definition of odd."},{"Start":"01:35.960 ","End":"01:37.610","Text":"Basically we\u0027re done,"},{"Start":"01:37.610 ","End":"01:42.990","Text":"except for those who want to know the reason why tangent of minus x is minus tangent x,"},{"Start":"01:42.990 ","End":"01:44.390","Text":"so let\u0027s check that."},{"Start":"01:44.390 ","End":"01:47.565","Text":"Tangent of minus x,"},{"Start":"01:47.565 ","End":"01:49.945","Text":"this is the optional part you can skip this,"},{"Start":"01:49.945 ","End":"01:58.114","Text":"is equal to sine of minus x over cosine of minus x,"},{"Start":"01:58.114 ","End":"02:00.425","Text":"because tangent is sine over cosine."},{"Start":"02:00.425 ","End":"02:03.290","Text":"Sine of minus x, by formula,"},{"Start":"02:03.290 ","End":"02:11.285","Text":"is the same as minus sine x. Cosine of minus x is the same as cosine x."},{"Start":"02:11.285 ","End":"02:14.375","Text":"Altogether, sine over cosine is tangent,"},{"Start":"02:14.375 ","End":"02:20.430","Text":"so we get minus tangent of x and we\u0027re done."}],"ID":4394},{"Watched":false,"Name":"Exercise 13","Duration":"1m 24s","ChapterTopicVideoID":4473,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this exercise, we have to say whether the function is even,"},{"Start":"00:02.640 ","End":"00:05.130","Text":"odd, or neither and to give reasons."},{"Start":"00:05.130 ","End":"00:10.065","Text":"The function here is given as y equals natural log of cosine x."},{"Start":"00:10.065 ","End":"00:13.620","Text":"Let\u0027s just write this in the functional form f(x)."},{"Start":"00:13.620 ","End":"00:15.960","Text":"I want to remind you what odd and even mean."},{"Start":"00:15.960 ","End":"00:24.170","Text":"That a function f is known as odd if f(-x) = - f(x),"},{"Start":"00:24.170 ","End":"00:31.140","Text":"and it\u0027s known as even if f(-x) = f(x) for all x."},{"Start":"00:31.140 ","End":"00:34.880","Text":"In our case, let\u0027s try putting x instead of -x."},{"Start":"00:34.880 ","End":"00:37.235","Text":"In other words, let\u0027s compute f(-x)."},{"Start":"00:37.235 ","End":"00:39.650","Text":"I won\u0027t get into the whole concept of domain of"},{"Start":"00:39.650 ","End":"00:44.195","Text":"definition here but let\u0027s assume that x is in the domain of definition."},{"Start":"00:44.195 ","End":"00:51.295","Text":"f(-x) is equal to the natural log of cosine of -x,"},{"Start":"00:51.295 ","End":"00:54.875","Text":"which is equal to the natural log of,"},{"Start":"00:54.875 ","End":"00:56.330","Text":"remembering our trigonometry,"},{"Start":"00:56.330 ","End":"00:59.900","Text":"cosine of -x is the same as cosine x."},{"Start":"00:59.900 ","End":"01:02.540","Text":"This brings us back to the original f(x),"},{"Start":"01:02.540 ","End":"01:05.450","Text":"so f(-x) is equal to f(x),"},{"Start":"01:05.450 ","End":"01:09.500","Text":"which fits the definition of even function."},{"Start":"01:09.500 ","End":"01:12.590","Text":"Basically we\u0027re done except that I want to mention that"},{"Start":"01:12.590 ","End":"01:16.025","Text":"we really should say what the domain is for this function."},{"Start":"01:16.025 ","End":"01:17.180","Text":"I\u0027ll just say a few words,"},{"Start":"01:17.180 ","End":"01:20.900","Text":"cosine x has to be positive because the natural log has to be positive,"},{"Start":"01:20.900 ","End":"01:25.619","Text":"and we can easily find out what cosine x is positive."}],"ID":4482},{"Watched":false,"Name":"Exercise 14","Duration":"3m 12s","ChapterTopicVideoID":4474,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"In this exercise,"},{"Start":"00:01.650 ","End":"00:04.955","Text":"we\u0027re given the graphs of 5 different functions."},{"Start":"00:04.955 ","End":"00:08.280","Text":"For each 1, we have to say whether the function is even,"},{"Start":"00:08.280 ","End":"00:11.400","Text":"odd or neither and to give reasons for our answer."},{"Start":"00:11.400 ","End":"00:15.435","Text":"Let\u0027s remember what odd and even mean graphically."},{"Start":"00:15.435 ","End":"00:20.700","Text":"An even function from the graph is one where if the point a,"},{"Start":"00:20.700 ","End":"00:22.289","Text":"b is on the graph,"},{"Start":"00:22.289 ","End":"00:24.660","Text":"then the point minus a,"},{"Start":"00:24.660 ","End":"00:27.015","Text":"b is also on the graph."},{"Start":"00:27.015 ","End":"00:30.915","Text":"This is also sometimes called symmetry about the y axis."},{"Start":"00:30.915 ","End":"00:32.580","Text":"If you put the y axis as a mirror,"},{"Start":"00:32.580 ","End":"00:34.260","Text":"you\u0027ll get the same thing."},{"Start":"00:34.260 ","End":"00:38.520","Text":"An odd function has the definition graphically that if a,"},{"Start":"00:38.520 ","End":"00:39.925","Text":"b is on the graph,"},{"Start":"00:39.925 ","End":"00:44.180","Text":"then minus a minus b is also on the graph."},{"Start":"00:44.180 ","End":"00:47.665","Text":"This is also called symmetry about the origin."},{"Start":"00:47.665 ","End":"00:49.985","Text":"Let\u0027s see in our 5 cases,"},{"Start":"00:49.985 ","End":"00:52.415","Text":"take each one and see what we have."},{"Start":"00:52.415 ","End":"00:54.125","Text":"In the first example,"},{"Start":"00:54.125 ","End":"00:56.825","Text":"it looks to me like it\u0027s an odd function."},{"Start":"00:56.825 ","End":"00:59.944","Text":"If I take a point a, b,"},{"Start":"00:59.944 ","End":"01:04.115","Text":"the point minus a minus b will be on the other side to the origin,"},{"Start":"01:04.115 ","End":"01:05.625","Text":"will be somewhere here."},{"Start":"01:05.625 ","End":"01:08.330","Text":"This is minus a, minus b."},{"Start":"01:08.330 ","End":"01:10.625","Text":"You can be for any a and b on this graph."},{"Start":"01:10.625 ","End":"01:12.230","Text":"This is odd."},{"Start":"01:12.230 ","End":"01:14.030","Text":"In the second example,"},{"Start":"01:14.030 ","End":"01:16.865","Text":"it looks to me like it\u0027s neither odd nor even."},{"Start":"01:16.865 ","End":"01:18.050","Text":"But to check this,"},{"Start":"01:18.050 ","End":"01:22.205","Text":"let\u0027s take this point for example,"},{"Start":"01:22.205 ","End":"01:23.885","Text":"I don\u0027t know its coordinates,"},{"Start":"01:23.885 ","End":"01:26.690","Text":"but let\u0027s just say that this was minus 1,"},{"Start":"01:26.690 ","End":"01:31.010","Text":"0, this is my a and this is my b."},{"Start":"01:31.010 ","End":"01:33.740","Text":"Now, if it was going to be even,"},{"Start":"01:33.740 ","End":"01:38.225","Text":"then minus a, b would have to be on the graph."},{"Start":"01:38.225 ","End":"01:40.370","Text":"Minus a, b would be 1,0,"},{"Start":"01:40.370 ","End":"01:41.930","Text":"which would be here,"},{"Start":"01:41.930 ","End":"01:43.475","Text":"which is not on the graph."},{"Start":"01:43.475 ","End":"01:48.125","Text":"If it was odd, I would need to look at minus a minus b,"},{"Start":"01:48.125 ","End":"01:51.215","Text":"which coincidentally also happens to be here,"},{"Start":"01:51.215 ","End":"01:53.450","Text":"and it\u0027s also not on the graph."},{"Start":"01:53.450 ","End":"01:56.830","Text":"This is neither odd nor even."},{"Start":"01:56.830 ","End":"01:58.515","Text":"In example c,"},{"Start":"01:58.515 ","End":"02:01.745","Text":"graphically I see a symmetry about the y-axis."},{"Start":"02:01.745 ","End":"02:03.635","Text":"But according to our definition,"},{"Start":"02:03.635 ","End":"02:06.890","Text":"let\u0027s just take a typical point here, a,"},{"Start":"02:06.890 ","End":"02:10.505","Text":"b and the corresponding point minus a,"},{"Start":"02:10.505 ","End":"02:12.440","Text":"b is on the graph."},{"Start":"02:12.440 ","End":"02:16.040","Text":"This tells us that this one is even."},{"Start":"02:16.040 ","End":"02:19.370","Text":"Now let\u0027s go to the next example."},{"Start":"02:19.370 ","End":"02:22.880","Text":"This looks to me like it\u0027s an odd function."},{"Start":"02:22.880 ","End":"02:24.425","Text":"Let\u0027s check this out."},{"Start":"02:24.425 ","End":"02:26.240","Text":"Let\u0027s take a typical point,"},{"Start":"02:26.240 ","End":"02:29.660","Text":"say here, and if this is a,"},{"Start":"02:29.660 ","End":"02:31.790","Text":"b, a corresponding point,"},{"Start":"02:31.790 ","End":"02:36.545","Text":"minus a minus b is simply the other side of the origin."},{"Start":"02:36.545 ","End":"02:38.825","Text":"That would be say here,"},{"Start":"02:38.825 ","End":"02:43.910","Text":"minus a, minus b. I could have taken any point along the graph."},{"Start":"02:43.910 ","End":"02:46.640","Text":"This function is odd."},{"Start":"02:46.640 ","End":"02:49.105","Text":"Next one e, also,"},{"Start":"02:49.105 ","End":"02:52.475","Text":"if I take any point here, a, b,"},{"Start":"02:52.475 ","End":"02:54.740","Text":"the point minus a minus b,"},{"Start":"02:54.740 ","End":"02:57.320","Text":"which is simply then the other side of the origin is"},{"Start":"02:57.320 ","End":"03:00.365","Text":"also on the graph and this could have been for any a, b."},{"Start":"03:00.365 ","End":"03:03.210","Text":"This function is also odd."},{"Start":"03:03.210 ","End":"03:05.715","Text":"In summary, a, e,"},{"Start":"03:05.715 ","End":"03:07.400","Text":"and d were odd,"},{"Start":"03:07.400 ","End":"03:12.990","Text":"c was even, and b was neither. We\u0027re done."}],"ID":4483},{"Watched":false,"Name":"Exercise 15","Duration":"5m 30s","ChapterTopicVideoID":4475,"CourseChapterTopicPlaylistID":1183,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.145","Text":"This question has several parts,"},{"Start":"00:02.145 ","End":"00:04.680","Text":"all involving odd and even functions."},{"Start":"00:04.680 ","End":"00:08.115","Text":"I\u0027d like to remind you what these terms mean."},{"Start":"00:08.115 ","End":"00:16.785","Text":"A function f is known as even if f of minus x equals f of x for all x,"},{"Start":"00:16.785 ","End":"00:24.870","Text":"and it\u0027s called odd if f of minus x is equal to minus f of x for all x."},{"Start":"00:24.870 ","End":"00:30.555","Text":"The first 2 questions involve the sum of 2 functions and in both these cases,"},{"Start":"00:30.555 ","End":"00:33.180","Text":"we\u0027ll let our function be f and"},{"Start":"00:33.180 ","End":"00:37.490","Text":"the other 2 functions will be g and h. In the first 2 questions,"},{"Start":"00:37.490 ","End":"00:43.610","Text":"we\u0027ll have f being the sum of g and h and in the other 3 functions,"},{"Start":"00:43.610 ","End":"00:46.170","Text":"we have a product each time."},{"Start":"00:46.170 ","End":"00:49.220","Text":"We\u0027ll let our function again be f,"},{"Start":"00:49.220 ","End":"00:54.740","Text":"but this time it\u0027s going to be the product of g of x and h of x."},{"Start":"00:54.740 ","End":"00:56.885","Text":"We\u0027ll do each piece separately."},{"Start":"00:56.885 ","End":"00:59.140","Text":"Let\u0027s start with a,"},{"Start":"00:59.140 ","End":"01:01.770","Text":"sum of 2 even functions is even."},{"Start":"01:01.770 ","End":"01:04.940","Text":"Let\u0027s compute f of minus x,"},{"Start":"01:04.940 ","End":"01:08.820","Text":"where f of x is given as g of x plus h of x."},{"Start":"01:08.820 ","End":"01:11.310","Text":"So f of minus x is equal,"},{"Start":"01:11.310 ","End":"01:19.114","Text":"we just substitute minus x equals g of minus x plus h of minus x."},{"Start":"01:19.114 ","End":"01:23.305","Text":"Now in a, both g and h are even,"},{"Start":"01:23.305 ","End":"01:29.495","Text":"so this equals g of x because g is even so g of minus x is g of x."},{"Start":"01:29.495 ","End":"01:31.505","Text":"Similarly for h, which is even so,"},{"Start":"01:31.505 ","End":"01:39.315","Text":"h of minus x equals h of x and g of x plus h of x is f of x by definition."},{"Start":"01:39.315 ","End":"01:46.610","Text":"What we get is that f of minus x is equal to f of x and that according to the definition,"},{"Start":"01:46.610 ","End":"01:50.660","Text":"makes f even, which is what we wanted to prove,"},{"Start":"01:50.660 ","End":"01:52.355","Text":"and we\u0027re done for part a."},{"Start":"01:52.355 ","End":"01:54.290","Text":"Now on to part b,"},{"Start":"01:54.290 ","End":"01:56.660","Text":"the sum of 2 odd functions is odd."},{"Start":"01:56.660 ","End":"02:00.410","Text":"Once again, our function will be f and it will be the sum of g and h,"},{"Start":"02:00.410 ","End":"02:01.960","Text":"both of which will be odd."},{"Start":"02:01.960 ","End":"02:04.820","Text":"Let\u0027s see, f of minus x,"},{"Start":"02:04.820 ","End":"02:12.820","Text":"since f is g plus h will equal g of minus x plus h of minus x."},{"Start":"02:12.820 ","End":"02:17.085","Text":"Part b, h and g are both odd because g is odd,"},{"Start":"02:17.085 ","End":"02:20.715","Text":"g of minus x is minus g of x."},{"Start":"02:20.715 ","End":"02:22.110","Text":"Because h is odd,"},{"Start":"02:22.110 ","End":"02:25.855","Text":"h of minus x is minus h of x."},{"Start":"02:25.855 ","End":"02:33.020","Text":"You can take the minus outside the brackets and we\u0027re left with g of x plus h of x."},{"Start":"02:33.020 ","End":"02:35.330","Text":"But g plus h is exactly f,"},{"Start":"02:35.330 ","End":"02:38.705","Text":"so this equals minus f of x."},{"Start":"02:38.705 ","End":"02:42.620","Text":"In other words, f of minus x equals minus f of"},{"Start":"02:42.620 ","End":"02:46.610","Text":"x and that fits the definition of an odd function,"},{"Start":"02:46.610 ","End":"02:48.980","Text":"which is what we were required to prove,"},{"Start":"02:48.980 ","End":"02:50.870","Text":"and we\u0027re done with part b."},{"Start":"02:50.870 ","End":"02:53.750","Text":"Next, we move on to part c. Here we"},{"Start":"02:53.750 ","End":"02:57.095","Text":"have the product of 2 even functions is an even function."},{"Start":"02:57.095 ","End":"03:04.505","Text":"For the product f is g times h. Let\u0027s see what we get for f of minus x."},{"Start":"03:04.505 ","End":"03:07.535","Text":"This equals since f is g times h,"},{"Start":"03:07.535 ","End":"03:12.800","Text":"g of minus x times h of minus x."},{"Start":"03:12.800 ","End":"03:16.835","Text":"But g and h here are both even functions,"},{"Start":"03:16.835 ","End":"03:21.480","Text":"so this equals g of x and h is also even,"},{"Start":"03:21.480 ","End":"03:24.454","Text":"so h of minus x is h of x,"},{"Start":"03:24.454 ","End":"03:27.680","Text":"but g of x, h of x is exactly f of x."},{"Start":"03:27.680 ","End":"03:31.820","Text":"In other words, f of minus x is equal to f of x,"},{"Start":"03:31.820 ","End":"03:34.550","Text":"which fits the definition of even,"},{"Start":"03:34.550 ","End":"03:40.580","Text":"which is what we needed to prove and we\u0027re done with part c. Next we have part d,"},{"Start":"03:40.580 ","End":"03:44.165","Text":"product of 2 odd functions is an even function."},{"Start":"03:44.165 ","End":"03:47.945","Text":"Our function f is equal to g times h. Let\u0027s see."},{"Start":"03:47.945 ","End":"03:51.110","Text":"What\u0027s f of minus x?"},{"Start":"03:51.110 ","End":"03:54.905","Text":"It\u0027s equal to g of minus x times h of minus x."},{"Start":"03:54.905 ","End":"03:58.075","Text":"In this case, we\u0027re given that g and h are both odd."},{"Start":"03:58.075 ","End":"04:06.560","Text":"G is odd, this gives us minus g of x times h of minus x will be minus h of x."},{"Start":"04:06.560 ","End":"04:09.955","Text":"This equals minus times minus is plus,"},{"Start":"04:09.955 ","End":"04:13.835","Text":"it\u0027s g of x times h of x."},{"Start":"04:13.835 ","End":"04:17.210","Text":"But this is just equal to f of x."},{"Start":"04:17.210 ","End":"04:21.875","Text":"In other words, f of minus x is equal to f of x."},{"Start":"04:21.875 ","End":"04:24.890","Text":"By definition, a function is even,"},{"Start":"04:24.890 ","End":"04:27.050","Text":"and this is what we were required to show."},{"Start":"04:27.050 ","End":"04:30.530","Text":"We\u0027re done with part d. Next is part e."},{"Start":"04:30.530 ","End":"04:34.834","Text":"The product of an even function and an odd function is an odd function."},{"Start":"04:34.834 ","End":"04:39.285","Text":"f is g times h. Let\u0027s take g as the even"},{"Start":"04:39.285 ","End":"04:44.250","Text":"and h as the odd and we have to show that f is odd."},{"Start":"04:44.250 ","End":"04:46.455","Text":"F of minus x,"},{"Start":"04:46.455 ","End":"04:48.240","Text":"since f is g times h,"},{"Start":"04:48.240 ","End":"04:53.280","Text":"is g of minus x times h of minus x."},{"Start":"04:53.280 ","End":"04:55.220","Text":"Now because g is even,"},{"Start":"04:55.220 ","End":"04:59.045","Text":"g of minus x is the same as g of x."},{"Start":"04:59.045 ","End":"05:01.325","Text":"But h being an odd function,"},{"Start":"05:01.325 ","End":"05:06.455","Text":"has the property that h of minus x is minus h of x."},{"Start":"05:06.455 ","End":"05:08.210","Text":"Bringing the minus upfront,"},{"Start":"05:08.210 ","End":"05:10.820","Text":"we get minus g of x,"},{"Start":"05:10.820 ","End":"05:15.620","Text":"h of x and this is exactly equal to minus f of x because f is"},{"Start":"05:15.620 ","End":"05:21.460","Text":"g times h. We have that f of minus x is minus f of x."},{"Start":"05:21.460 ","End":"05:26.160","Text":"This fits the definition of an odd function and this is what we had to show."},{"Start":"05:26.160 ","End":"05:30.310","Text":"We\u0027re done with part e and with the whole exercise."}],"ID":4484}],"Thumbnail":null,"ID":1183},{"Name":"One-to-One Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"One-to-One Functions","Duration":"12m 19s","ChapterTopicVideoID":9302,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this clip, I\u0027m going to explain the concept of a 1-to-1 function."},{"Start":"00:04.320 ","End":"00:10.950","Text":"I\u0027d like to start right away with an example of a well-known function, f(x) = x^2."},{"Start":"00:10.950 ","End":"00:12.225","Text":"Here\u0027s a rough sketch,"},{"Start":"00:12.225 ","End":"00:16.319","Text":"and now I\u0027d like to examine 2 different values of x."},{"Start":"00:16.319 ","End":"00:20.380","Text":"In this case, I\u0027d like to look at x = 2"},{"Start":"00:20.380 ","End":"00:25.680","Text":"and x = minus 2 and see what happens to the values of y."},{"Start":"00:25.680 ","End":"00:26.970","Text":"Well, if x is 2,"},{"Start":"00:26.970 ","End":"00:29.865","Text":"y = 2^2, which is 4."},{"Start":"00:29.865 ","End":"00:32.415","Text":"If I put x = minus 2,"},{"Start":"00:32.415 ","End":"00:34.725","Text":"I get f(x) is minus 2^2,"},{"Start":"00:34.725 ","End":"00:36.315","Text":"which is also 4."},{"Start":"00:36.315 ","End":"00:40.560","Text":"So notice that these 2 fall on the y = 4 line,"},{"Start":"00:40.560 ","End":"00:42.410","Text":"but the main thing I want to observe is that"},{"Start":"00:42.410 ","End":"00:47.735","Text":"2 different values of x went to the same value of y,"},{"Start":"00:47.735 ","End":"00:51.900","Text":"f(2) = f(-2),"},{"Start":"00:51.900 ","End":"00:53.745","Text":"they\u0027re both equal to 4."},{"Start":"00:53.745 ","End":"00:59.625","Text":"In general, I would write this f(x_1) = f(x_2),"},{"Start":"00:59.625 ","End":"01:04.035","Text":"even though x_1 is not equal to x_2,"},{"Start":"01:04.035 ","End":"01:05.880","Text":"like this minus 2 and 2."},{"Start":"01:05.880 ","End":"01:07.325","Text":"So these 2 were different."},{"Start":"01:07.325 ","End":"01:09.785","Text":"However, when I substituted them,"},{"Start":"01:09.785 ","End":"01:11.845","Text":"I got an inequality."},{"Start":"01:11.845 ","End":"01:13.425","Text":"With this same function,"},{"Start":"01:13.425 ","End":"01:15.720","Text":"I could put 2 different values of x,"},{"Start":"01:15.720 ","End":"01:17.865","Text":"1 and 2, and in this case,"},{"Start":"01:17.865 ","End":"01:25.185","Text":"they are different and the y\u0027s are also different because f(1) is 1 and f(2) is 4."},{"Start":"01:25.185 ","End":"01:30.200","Text":"Sometimes you could have 2 different values of x that give you the same value of y,"},{"Start":"01:30.200 ","End":"01:34.385","Text":"but mostly 2 different values of x will give you different values of y."},{"Start":"01:34.385 ","End":"01:38.605","Text":"Let\u0027s contrast this with another function where f(x) = x."},{"Start":"01:38.605 ","End":"01:43.240","Text":"Here I sketched a function f(x) = x."},{"Start":"01:43.240 ","End":"01:47.030","Text":"In this case, if I try substituting 2 different values of x,"},{"Start":"01:47.030 ","End":"01:50.195","Text":"let\u0027s say I use the same 1s minus 2 and 2,"},{"Start":"01:50.195 ","End":"01:52.465","Text":"I get different values of y."},{"Start":"01:52.465 ","End":"01:55.425","Text":"2 gives me 2, minus 2 gives me minus 2."},{"Start":"01:55.425 ","End":"01:58.905","Text":"Every time x_1 is different from x_2,"},{"Start":"01:58.905 ","End":"02:03.570","Text":"f(x_1) will be different from f(x_2)."},{"Start":"02:03.570 ","End":"02:05.865","Text":"I write this arrow as implies,"},{"Start":"02:05.865 ","End":"02:10.025","Text":"I mean whenever I take 2 different values of x,"},{"Start":"02:10.025 ","End":"02:11.210","Text":"whatever those 2 are,"},{"Start":"02:11.210 ","End":"02:12.890","Text":"as long as they\u0027re different,"},{"Start":"02:12.890 ","End":"02:15.745","Text":"f(x_1) will be different from f(x_2)."},{"Start":"02:15.745 ","End":"02:19.950","Text":"Here, I write this arrow like a but, or, and."},{"Start":"02:19.950 ","End":"02:21.230","Text":"Either thing could happen."},{"Start":"02:21.230 ","End":"02:24.800","Text":"You could have different values of x like 2 and 3,"},{"Start":"02:24.800 ","End":"02:25.940","Text":"and if you substitute,"},{"Start":"02:25.940 ","End":"02:28.415","Text":"you\u0027ll see you\u0027ll get 2 different values of f(x)."},{"Start":"02:28.415 ","End":"02:32.190","Text":"But sometimes you can take 2 different values of x and get"},{"Start":"02:32.190 ","End":"02:36.964","Text":"the same value of f(x) and that\u0027s the big difference between these 2 functions."},{"Start":"02:36.964 ","End":"02:41.540","Text":"A function where this holds that different values of x always go to"},{"Start":"02:41.540 ","End":"02:46.650","Text":"different values of y is called a 1-to-1 function,"},{"Start":"02:46.650 ","End":"02:47.790","Text":"and in this case,"},{"Start":"02:47.790 ","End":"02:54.040","Text":"the straight line f(x) = x or y = x is an example of a 1-to-1 function."},{"Start":"02:54.040 ","End":"02:56.100","Text":"I haven\u0027t exactly proven it to you,"},{"Start":"02:56.100 ","End":"02:59.160","Text":"but it\u0027s quite clear that because f(x) is equal to x,"},{"Start":"02:59.160 ","End":"03:02.310","Text":"if I take 2 different values of x like 2 and 3,"},{"Start":"03:02.310 ","End":"03:04.290","Text":"now I\u0027ll get the y\u0027s are the same,"},{"Start":"03:04.290 ","End":"03:05.370","Text":"they\u0027ll also be 2 and 3,"},{"Start":"03:05.370 ","End":"03:06.710","Text":"so they\u0027ll also be different."},{"Start":"03:06.710 ","End":"03:10.025","Text":"Now, in order for something not to be 1-to-1,"},{"Start":"03:10.025 ","End":"03:12.455","Text":"all I need is 1 single example."},{"Start":"03:12.455 ","End":"03:18.770","Text":"So the opposite of different values of x go to different values of y is that there"},{"Start":"03:18.770 ","End":"03:25.355","Text":"is at least 1 pair of x that are different and yet f(x_1) and f(x_2) are the same."},{"Start":"03:25.355 ","End":"03:28.360","Text":"So this is not 1-to-1,"},{"Start":"03:28.360 ","End":"03:31.370","Text":"and not 1-to-1 is generally easy to prove,"},{"Start":"03:31.370 ","End":"03:34.370","Text":"as I say, because all you need is to find a single example of"},{"Start":"03:34.370 ","End":"03:37.745","Text":"a pair of x\u0027s which are different but go to the same y."},{"Start":"03:37.745 ","End":"03:39.620","Text":"The 1-to-1, generally,"},{"Start":"03:39.620 ","End":"03:41.315","Text":"you need a proper proof,"},{"Start":"03:41.315 ","End":"03:42.815","Text":"but don\u0027t be frightened."},{"Start":"03:42.815 ","End":"03:45.860","Text":"I\u0027ll show you the example of this kind of proof is easy,"},{"Start":"03:45.860 ","End":"03:47.599","Text":"is to prove that in general,"},{"Start":"03:47.599 ","End":"03:49.370","Text":"if you take 2 different values of x,"},{"Start":"03:49.370 ","End":"03:51.365","Text":"you go to different values of y."},{"Start":"03:51.365 ","End":"03:53.960","Text":"Before we get onto the exercises,"},{"Start":"03:53.960 ","End":"03:56.480","Text":"I\u0027d like to rephrase the definition of"},{"Start":"03:56.480 ","End":"04:02.120","Text":"a 1-to-1 function and also to give an equivalent definition which is easier to work with."},{"Start":"04:02.120 ","End":"04:07.055","Text":"Now the definition we gave from earlier was in this highlight in yellow here,"},{"Start":"04:07.055 ","End":"04:11.630","Text":"1-to-1 means that if x_1 is not equal to x_2,"},{"Start":"04:11.630 ","End":"04:14.860","Text":"then f(x_1) is not equal to f(x_2)."},{"Start":"04:14.860 ","End":"04:19.790","Text":"Somehow the x_1, x_2 seems unfriendly and I\u0027ve replaced that with a and b."},{"Start":"04:19.790 ","End":"04:22.580","Text":"What we get is if a is not equal to b,"},{"Start":"04:22.580 ","End":"04:25.190","Text":"then f(a) is not equal to f(b)."},{"Start":"04:25.190 ","End":"04:29.945","Text":"In other words, different values of x go to different values of y."},{"Start":"04:29.945 ","End":"04:32.495","Text":"Now a more practical definition,"},{"Start":"04:32.495 ","End":"04:36.775","Text":"which is actually equivalent logically to this 1 is to say the following,"},{"Start":"04:36.775 ","End":"04:40.350","Text":"that if f(a) is equal to f(b),"},{"Start":"04:40.350 ","End":"04:42.335","Text":"then a = b."},{"Start":"04:42.335 ","End":"04:45.590","Text":"In other words, if 2 x\u0027s are taken by f to the same y,"},{"Start":"04:45.590 ","End":"04:48.305","Text":"then from the beginning, they were the same x."},{"Start":"04:48.305 ","End":"04:51.380","Text":"In logic, this is called the contrapositive of"},{"Start":"04:51.380 ","End":"04:54.710","Text":"this 1 where you reverse the arrow and you negate the statement."},{"Start":"04:54.710 ","End":"04:56.630","Text":"I\u0027ll see if I can explain it in words."},{"Start":"04:56.630 ","End":"04:59.460","Text":"If different things go to different things,"},{"Start":"04:59.460 ","End":"05:02.915","Text":"things that go to the same thing must be the same."},{"Start":"05:02.915 ","End":"05:05.345","Text":"If you don\u0027t follow the logic, not to worry."},{"Start":"05:05.345 ","End":"05:09.945","Text":"We\u0027ll just take this as our new definition of 1-to-1."},{"Start":"05:09.945 ","End":"05:15.980","Text":"This is the definition we\u0027ll be using when proving that a function is 1-to-1."},{"Start":"05:15.980 ","End":"05:17.465","Text":"Now in a typical exercise,"},{"Start":"05:17.465 ","End":"05:20.810","Text":"you\u0027re given a function and asked whether it\u0027s 1-to-1 or not."},{"Start":"05:20.810 ","End":"05:23.570","Text":"You can make a guess, but usually you start off with trying"},{"Start":"05:23.570 ","End":"05:26.240","Text":"to prove that it is 1-to-1 using this,"},{"Start":"05:26.240 ","End":"05:27.560","Text":"but if that doesn\u0027t work,"},{"Start":"05:27.560 ","End":"05:30.440","Text":"then you try to prove that it\u0027s not 1-to-1."},{"Start":"05:30.440 ","End":"05:34.400","Text":"Now, that\u0027s much easier because to disprove this thing,"},{"Start":"05:34.400 ","End":"05:38.100","Text":"all I have to do is produce a single example, in other words,"},{"Start":"05:38.100 ","End":"05:39.355","Text":"a pair of numbers,"},{"Start":"05:39.355 ","End":"05:41.495","Text":"a and b, which are different,"},{"Start":"05:41.495 ","End":"05:46.385","Text":"but nevertheless they go to the same y just as we did above."},{"Start":"05:46.385 ","End":"05:48.565","Text":"We found 2 and minus 2,"},{"Start":"05:48.565 ","End":"05:49.965","Text":"which were different,"},{"Start":"05:49.965 ","End":"05:52.980","Text":"and yet f took them both to the same place,"},{"Start":"05:52.980 ","End":"05:54.135","Text":"to the same y."},{"Start":"05:54.135 ","End":"05:57.845","Text":"This is used for proving and this is used for disproving."},{"Start":"05:57.845 ","End":"06:01.475","Text":"Now let\u0027s get on to some exercises to make this clearer."},{"Start":"06:01.475 ","End":"06:06.765","Text":"Given that f(x) = 4x plus 10,"},{"Start":"06:06.765 ","End":"06:10.159","Text":"either to prove that it\u0027s 1-to-1 or disprove it."},{"Start":"06:10.159 ","End":"06:13.025","Text":"We usually try to prove that it is 1-to-1,"},{"Start":"06:13.025 ","End":"06:17.870","Text":"and so we start with this statement f(a) = f(b) and try to work our way"},{"Start":"06:17.870 ","End":"06:22.775","Text":"through a series of algebraic steps or otherwise and reach a = b at the end."},{"Start":"06:22.775 ","End":"06:28.885","Text":"Let\u0027s start off with just writing f(a) = f(b)."},{"Start":"06:28.885 ","End":"06:31.772","Text":"Now I translate this because I know what f is,"},{"Start":"06:31.772 ","End":"06:33.980","Text":"f is given by this formula."},{"Start":"06:33.980 ","End":"06:39.265","Text":"F(a) is 4a plus 10 and this = 4b plus 10."},{"Start":"06:39.265 ","End":"06:43.595","Text":"From here, the obvious thing to do is to subtract 10 from both sides,"},{"Start":"06:43.595 ","End":"06:47.345","Text":"so we get 4a = 4b."},{"Start":"06:47.345 ","End":"06:50.770","Text":"Then divide both sides by 4,"},{"Start":"06:50.770 ","End":"06:53.715","Text":"and so we\u0027re left with a = b."},{"Start":"06:53.715 ","End":"06:56.790","Text":"This statement led to this statement,"},{"Start":"06:56.790 ","End":"07:00.860","Text":"so we showed indeed that this holds and that\u0027s all there is to it,"},{"Start":"07:00.860 ","End":"07:02.725","Text":"so this is 1-to-1."},{"Start":"07:02.725 ","End":"07:13.135","Text":"Our next exercise is f(x) = x plus 1/x plus 4."},{"Start":"07:13.135 ","End":"07:17.540","Text":"Once again, we\u0027re going to start off by trying to prove that it is 1-to-1."},{"Start":"07:17.540 ","End":"07:19.760","Text":"So f(a) = f(b)."},{"Start":"07:19.760 ","End":"07:21.545","Text":"That\u0027s the first thing we write."},{"Start":"07:21.545 ","End":"07:24.605","Text":"To translate this according to the definition of f,"},{"Start":"07:24.605 ","End":"07:34.855","Text":"and I\u0027ll get that a plus 1/a plus 4 = b plus 1/b plus 4."},{"Start":"07:34.855 ","End":"07:38.660","Text":"In algebra, the usual thing to do is to cross multiply."},{"Start":"07:38.660 ","End":"07:41.395","Text":"This times this equals this times this."},{"Start":"07:41.395 ","End":"07:46.620","Text":"A plus 1 times b plus 4"},{"Start":"07:46.620 ","End":"07:53.055","Text":"= a plus 4 times b plus 1."},{"Start":"07:53.055 ","End":"07:56.730","Text":"Then open up the brackets and now look,"},{"Start":"07:56.730 ","End":"07:58.485","Text":"a lot of stuff cancels,"},{"Start":"07:58.485 ","End":"08:04.800","Text":"like ab cancels with ab and 4 with 4."},{"Start":"08:04.800 ","End":"08:09.180","Text":"We will get 3a = 3b,"},{"Start":"08:09.180 ","End":"08:12.495","Text":"therefore a = b,"},{"Start":"08:12.495 ","End":"08:15.180","Text":"and so this is, yes,"},{"Start":"08:15.180 ","End":"08:19.490","Text":"it is 1-to-1 because we managed to get there,"},{"Start":"08:19.490 ","End":"08:21.080","Text":"to the a = b."},{"Start":"08:21.080 ","End":"08:22.925","Text":"We should, in our last example,"},{"Start":"08:22.925 ","End":"08:25.460","Text":"take an example where it isn\u0027t 1-to-1."},{"Start":"08:25.460 ","End":"08:28.310","Text":"In fact, we already encountered such an example above."},{"Start":"08:28.310 ","End":"08:31.655","Text":"We had that f(x) = x^2,"},{"Start":"08:31.655 ","End":"08:37.575","Text":"and remember we showed that if we put in x = 2 or x = minus 2,"},{"Start":"08:37.575 ","End":"08:39.135","Text":"we got the same thing."},{"Start":"08:39.135 ","End":"08:41.180","Text":"But let\u0027s say we didn\u0027t remember that,"},{"Start":"08:41.180 ","End":"08:42.290","Text":"we didn\u0027t know about that."},{"Start":"08:42.290 ","End":"08:44.135","Text":"This is a fresh exercise,"},{"Start":"08:44.135 ","End":"08:50.210","Text":"so we start off with the statement f(a) = f(b)."},{"Start":"08:50.210 ","End":"08:53.240","Text":"Try and work our way down to a = b."},{"Start":"08:53.240 ","End":"08:54.500","Text":"Let\u0027s see what happens."},{"Start":"08:54.500 ","End":"08:56.180","Text":"We get that a^2,"},{"Start":"08:56.180 ","End":"09:01.075","Text":"which is f(a) = b^2, which is f(b)."},{"Start":"09:01.075 ","End":"09:06.470","Text":"From here we can see that either a = b or a = minus b,"},{"Start":"09:06.470 ","End":"09:07.770","Text":"it should be clear enough."},{"Start":"09:07.770 ","End":"09:10.190","Text":"We didn\u0027t always get a = b."},{"Start":"09:10.190 ","End":"09:13.480","Text":"We also got the possibility a = minus b,"},{"Start":"09:13.480 ","End":"09:17.450","Text":"and that in fact reflects in the example we had above where we substituted"},{"Start":"09:17.450 ","End":"09:21.765","Text":"x = 2 or minus 2 and we got the same answer."},{"Start":"09:21.765 ","End":"09:23.750","Text":"Now we can\u0027t just leave it like that."},{"Start":"09:23.750 ","End":"09:27.350","Text":"Strictly speaking, in order to prove that something is not 1-to-1,"},{"Start":"09:27.350 ","End":"09:29.120","Text":"we have to find a and b."},{"Start":"09:29.120 ","End":"09:31.445","Text":"Let\u0027s just reuse the ones we had before."},{"Start":"09:31.445 ","End":"09:35.955","Text":"We say, let\u0027s take a= 2,"},{"Start":"09:35.955 ","End":"09:39.050","Text":"b=minus 2, or the other way around."},{"Start":"09:39.050 ","End":"09:40.175","Text":"It doesn\u0027t matter."},{"Start":"09:40.175 ","End":"09:41.885","Text":"We found our a and b,"},{"Start":"09:41.885 ","End":"09:44.135","Text":"a is not equal to b."},{"Start":"09:44.135 ","End":"09:52.165","Text":"Look, a is not equal to b because 2 is not equal to minus 2, but f(a),"},{"Start":"09:52.165 ","End":"09:55.560","Text":"which is equal to 2^2 is 4,"},{"Start":"09:55.560 ","End":"10:00.120","Text":"is actually equal to f(b),"},{"Start":"10:00.120 ","End":"10:02.715","Text":"which is also f of minus 2,"},{"Start":"10:02.715 ","End":"10:04.170","Text":"which is also 4."},{"Start":"10:04.170 ","End":"10:07.275","Text":"We proved this. We found a not equal to b,"},{"Start":"10:07.275 ","End":"10:10.995","Text":"and if I cut out the middleman, f(a) = f(b)."},{"Start":"10:10.995 ","End":"10:15.835","Text":"In this case, this 1 is not 1-to-1."},{"Start":"10:15.835 ","End":"10:20.030","Text":"Finally, I\u0027d like to show how all this works graphically."},{"Start":"10:20.030 ","End":"10:25.610","Text":"I would like to start with the example of a function which is not 1-to-1."},{"Start":"10:25.610 ","End":"10:28.175","Text":"Up above, we had y = x^2."},{"Start":"10:28.175 ","End":"10:33.795","Text":"Up there we had 2 and minus 2 both go to 4,"},{"Start":"10:33.795 ","End":"10:38.960","Text":"so (2, 4) is on the graph and so as minus 2, 4."},{"Start":"10:38.960 ","End":"10:43.955","Text":"We even mentioned that if you draw the horizontal line y = 4,"},{"Start":"10:43.955 ","End":"10:46.465","Text":"that it goes through both points."},{"Start":"10:46.465 ","End":"10:48.440","Text":"What we see is that if we have"},{"Start":"10:48.440 ","End":"10:53.635","Text":"a horizontal line which cuts the graph at 2 different points,"},{"Start":"10:53.635 ","End":"10:55.845","Text":"then the function is not 1-to-1,"},{"Start":"10:55.845 ","End":"10:59.555","Text":"because this horizontal line is the line y = something,"},{"Start":"10:59.555 ","End":"11:02.180","Text":"in our case 4, and if it cuts at 2 points,"},{"Start":"11:02.180 ","End":"11:05.990","Text":"then these 2 points are the 2 x\u0027s or the a and the b,"},{"Start":"11:05.990 ","End":"11:07.745","Text":"say this is a and this is b,"},{"Start":"11:07.745 ","End":"11:10.070","Text":"where f(a) and f(b) both go to"},{"Start":"11:10.070 ","End":"11:13.940","Text":"the same point because they\u0027re on the same horizontal line."},{"Start":"11:13.940 ","End":"11:16.055","Text":"The y of both is 4."},{"Start":"11:16.055 ","End":"11:18.290","Text":"Conversely, if I have a graph,"},{"Start":"11:18.290 ","End":"11:23.600","Text":"then if we take a ruler or something and draw horizontal lines,"},{"Start":"11:23.600 ","End":"11:27.664","Text":"there is no place where a horizontal line cuts the function twice."},{"Start":"11:27.664 ","End":"11:29.375","Text":"I can actually even tell you more than that."},{"Start":"11:29.375 ","End":"11:32.015","Text":"If a function is increasing or decreasing,"},{"Start":"11:32.015 ","End":"11:34.505","Text":"then horizontal line can\u0027t cut it twice,"},{"Start":"11:34.505 ","End":"11:37.470","Text":"because every time you move along it gets higher."},{"Start":"11:37.470 ","End":"11:40.935","Text":"In this case, horizontal line can\u0027t cut it twice,"},{"Start":"11:40.935 ","End":"11:42.870","Text":"so it is 1-to-1."},{"Start":"11:42.870 ","End":"11:47.845","Text":"This one is not 1-to-1 because it has a horizontal line that cuts it twice,"},{"Start":"11:47.845 ","End":"11:51.785","Text":"and this one is 1-to-1 because it doesn\u0027t cut it twice."},{"Start":"11:51.785 ","End":"11:54.650","Text":"Of course, it could cut it any number of times."},{"Start":"11:54.650 ","End":"11:57.380","Text":"I could have a function that looks like this,"},{"Start":"11:57.380 ","End":"12:05.665","Text":"and in this case I could get a horizontal line that would cut it even at 3 points here,"},{"Start":"12:05.665 ","End":"12:07.920","Text":"here, and here,"},{"Start":"12:07.920 ","End":"12:13.065","Text":"and f(a) and f(b) and f(c) would all be the same."},{"Start":"12:13.065 ","End":"12:15.895","Text":"This is also certainly not 1-to-1."},{"Start":"12:15.895 ","End":"12:19.980","Text":"I think this concludes the clip on 1-to-1."}],"ID":9614},{"Watched":false,"Name":"Exercise 1","Duration":"1m 55s","ChapterTopicVideoID":4384,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we\u0027re given the graphs of"},{"Start":"00:02.550 ","End":"00:06.795","Text":"several functions and we have to say which of these functions is 1:1."},{"Start":"00:06.795 ","End":"00:10.230","Text":"Now remember, a function is 1:1 if"},{"Start":"00:10.230 ","End":"00:14.775","Text":"a horizontal line can cross the function at most one time."},{"Start":"00:14.775 ","End":"00:16.920","Text":"Let\u0027s look at both A and B."},{"Start":"00:16.920 ","End":"00:22.770","Text":"In A, we see that any horizontal line will only cut the function at most one time."},{"Start":"00:22.770 ","End":"00:25.215","Text":"It can\u0027t cut twice, but in B,"},{"Start":"00:25.215 ","End":"00:28.725","Text":"a horizontal line can cut the function twice,"},{"Start":"00:28.725 ","End":"00:31.260","Text":"for example, here and here."},{"Start":"00:31.260 ","End":"00:33.435","Text":"For example, if this was say,"},{"Start":"00:33.435 ","End":"00:38.415","Text":"minus 2 and this was 2 and this value was 1,"},{"Start":"00:38.415 ","End":"00:44.105","Text":"then we would have that f of minus 2 equals f(2),"},{"Start":"00:44.105 ","End":"00:46.880","Text":"but minus 2 is not equal 2,"},{"Start":"00:46.880 ","End":"00:50.450","Text":"so it\u0027s not 1:1 on the algebraic definition also."},{"Start":"00:50.450 ","End":"00:54.570","Text":"In other words, so far we have that A, we can say,"},{"Start":"00:54.570 ","End":"00:57.195","Text":"yes, it is 1:1,"},{"Start":"00:57.195 ","End":"00:59.490","Text":"and in B the answer is no."},{"Start":"00:59.490 ","End":"01:01.880","Text":"Continuing to C and D,"},{"Start":"01:01.880 ","End":"01:04.880","Text":"it\u0027s clear that neither of these is 1;1 because,"},{"Start":"01:04.880 ","End":"01:09.510","Text":"for example, a horizontal line here will cross twice."},{"Start":"01:09.510 ","End":"01:12.620","Text":"Similarly here, not only connect cross twice,"},{"Start":"01:12.620 ","End":"01:18.595","Text":"horizontal line will cut many times here and so on."},{"Start":"01:18.595 ","End":"01:21.990","Text":"For these we can answer that this is no,"},{"Start":"01:21.990 ","End":"01:25.280","Text":"not 1:1, and also here, it\u0027s no."},{"Start":"01:25.280 ","End":"01:26.750","Text":"In example E,"},{"Start":"01:26.750 ","End":"01:33.800","Text":"we also see that it is 1:1 because any horizontal line will cut at most once."},{"Start":"01:33.800 ","End":"01:37.070","Text":"Of course, a horizontal line might not cut at all"},{"Start":"01:37.070 ","End":"01:40.795","Text":"but the point is that it must not cut more than once."},{"Start":"01:40.795 ","End":"01:43.190","Text":"So E is definitely 1:1."},{"Start":"01:43.190 ","End":"01:45.505","Text":"So is F,"},{"Start":"01:45.505 ","End":"01:50.300","Text":"any horizontal line that we control will not cut the graph twice."},{"Start":"01:50.300 ","End":"01:52.865","Text":"F is also 1:1,"},{"Start":"01:52.865 ","End":"01:55.800","Text":"and we\u0027re done for this exercise."}],"ID":4393},{"Watched":false,"Name":"Exercise 2","Duration":"2m 17s","ChapterTopicVideoID":4386,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we have to find the inverse function of this function f of x,"},{"Start":"00:04.770 ","End":"00:07.665","Text":"which is log to the base 2 of this expression."},{"Start":"00:07.665 ","End":"00:10.695","Text":"The domain is x bigger than 1."},{"Start":"00:10.695 ","End":"00:12.660","Text":"When x is bigger than 1,"},{"Start":"00:12.660 ","End":"00:15.960","Text":"certainly not 0 so 1 over x is okay."},{"Start":"00:15.960 ","End":"00:18.540","Text":"Also this thing will be positive."},{"Start":"00:18.540 ","End":"00:20.880","Text":"I could have even written x bigger than 0 and it"},{"Start":"00:20.880 ","End":"00:23.475","Text":"would still be positive and everything would be okay."},{"Start":"00:23.475 ","End":"00:27.720","Text":"Anyway let y equal f of x so y is equal to this."},{"Start":"00:27.720 ","End":"00:30.870","Text":"Now to find the inverse function is one way of doing it"},{"Start":"00:30.870 ","End":"00:34.210","Text":"is to get x in terms of y step-by-step."},{"Start":"00:34.210 ","End":"00:36.600","Text":"Let\u0027s first of all get rid of the log to the base 2."},{"Start":"00:36.600 ","End":"00:39.430","Text":"This is what we get just using the rules of logarithms."},{"Start":"00:39.430 ","End":"00:41.320","Text":"The log to the base 2 of this is this,"},{"Start":"00:41.320 ","End":"00:43.225","Text":"then 2 to this is this."},{"Start":"00:43.225 ","End":"00:45.830","Text":"Now we want to get rid of the denominator here,"},{"Start":"00:45.830 ","End":"00:48.490","Text":"multiply everything by x, and I get this."},{"Start":"00:48.490 ","End":"00:51.635","Text":"You can see it\u0027s a quadratic equation in x,"},{"Start":"00:51.635 ","End":"00:53.950","Text":"especially when I rewrite it like this."},{"Start":"00:53.950 ","End":"00:58.355","Text":"I\u0027m going to solve this quadratic equation using the quadratic formula."},{"Start":"00:58.355 ","End":"01:00.020","Text":"This is what the formula gives."},{"Start":"01:00.020 ","End":"01:02.765","Text":"The trouble is that there\u0027s x1 and x2,"},{"Start":"01:02.765 ","End":"01:05.000","Text":"we\u0027ll we can take the plus and we can take the minus,"},{"Start":"01:05.000 ","End":"01:06.515","Text":"and that wouldn\u0027t be a function."},{"Start":"01:06.515 ","End":"01:09.650","Text":"However, because x is bigger than 1,"},{"Start":"01:09.650 ","End":"01:12.215","Text":"it turns out that only the plus is a solution."},{"Start":"01:12.215 ","End":"01:14.240","Text":"I\u0027m not going to go into all the algebra,"},{"Start":"01:14.240 ","End":"01:16.550","Text":"but if we take the minus is not going to be bigger than"},{"Start":"01:16.550 ","End":"01:19.760","Text":"1 and so we now have x in terms of y."},{"Start":"01:19.760 ","End":"01:22.430","Text":"Now, this actually gives us the inverse function,"},{"Start":"01:22.430 ","End":"01:23.960","Text":"but not quite in the way we want."},{"Start":"01:23.960 ","End":"01:27.695","Text":"It gives us x is the inverse function of y,"},{"Start":"01:27.695 ","End":"01:31.595","Text":"but we want to write the inverse function has y in terms of x."},{"Start":"01:31.595 ","End":"01:33.020","Text":"The latter is a dummy variable,"},{"Start":"01:33.020 ","End":"01:35.120","Text":"so I just switched the x and the y."},{"Start":"01:35.120 ","End":"01:36.260","Text":"Perhaps I will write that here."},{"Start":"01:36.260 ","End":"01:38.720","Text":"Yeah, we switched x and y to get"},{"Start":"01:38.720 ","End":"01:42.230","Text":"the function because the function doesn\u0027t depend on the variable."},{"Start":"01:42.230 ","End":"01:44.990","Text":"This is x as a function of y so this is the function of x."},{"Start":"01:44.990 ","End":"01:46.790","Text":"You replace y by x."},{"Start":"01:46.790 ","End":"01:51.769","Text":"The last thing I want us to relate to the domain of the function,"},{"Start":"01:51.769 ","End":"01:54.560","Text":"when is this f to the minus 1 defined?"},{"Start":"01:54.560 ","End":"01:58.835","Text":"Well, what\u0027s under the square root sign has to be non-negative."},{"Start":"01:58.835 ","End":"02:03.005","Text":"This what\u0027s under the square root has to be bigger or equal to 0 for the domain."},{"Start":"02:03.005 ","End":"02:04.400","Text":"This is not hard to solve,"},{"Start":"02:04.400 ","End":"02:08.870","Text":"just bring the 4 to the other side and notice that 4 is 2 squared."},{"Start":"02:08.870 ","End":"02:11.570","Text":"Now I have 2 things equal with equal basis."},{"Start":"02:11.570 ","End":"02:13.875","Text":"There\u0027s a 2 here and a 2 here."},{"Start":"02:13.875 ","End":"02:16.480","Text":"Since the base is bigger than 1,"},{"Start":"02:16.480 ","End":"02:20.270","Text":"it\u0027s an increasing function so 2x is bigger than 2,"},{"Start":"02:20.270 ","End":"02:22.715","Text":"and finally x bigger or equal to 1."},{"Start":"02:22.715 ","End":"02:29.910","Text":"To summarize, we found the inverse function here and its domain is here and we\u0027re done."}],"ID":4395},{"Watched":false,"Name":"Exercise 3","Duration":"5m 45s","ChapterTopicVideoID":4839,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we\u0027re given 3 functions here, a, b,"},{"Start":"00:03.510 ","End":"00:04.740","Text":"and c. In each case,"},{"Start":"00:04.740 ","End":"00:07.650","Text":"we have to see whether the function is 1-1."},{"Start":"00:07.650 ","End":"00:11.295","Text":"I\u0027d like to remind you what it means to be 1-1."},{"Start":"00:11.295 ","End":"00:14.640","Text":"If we have a function f,"},{"Start":"00:14.640 ","End":"00:20.460","Text":"and if whenever we have that f of a equals f of b,"},{"Start":"00:20.460 ","End":"00:26.160","Text":"that it automatically imply that in every case that a equals b."},{"Start":"00:26.160 ","End":"00:31.920","Text":"In other words, you can always show maybe by a series of steps."},{"Start":"00:31.920 ","End":"00:35.520","Text":"You start off from f of a equals f of b and end up with a equals b."},{"Start":"00:35.520 ","End":"00:36.720","Text":"If this is so always,"},{"Start":"00:36.720 ","End":"00:39.090","Text":"then the function is 1-1."},{"Start":"00:39.090 ","End":"00:42.450","Text":"Let\u0027s go for part a first."},{"Start":"00:42.450 ","End":"00:47.940","Text":"We start off with saying f of a equals f of b."},{"Start":"00:48.190 ","End":"00:52.520","Text":"We want to end up at the end with a equals b."},{"Start":"00:52.520 ","End":"00:57.815","Text":"Assume a and b are in the domain of definition but the denominator is not 0."},{"Start":"00:57.815 ","End":"01:02.495","Text":"If we put a in the left-hand side, plug it in,"},{"Start":"01:02.495 ","End":"01:10.670","Text":"we get 2a plus 1 over 4a minus 2 and f of b is the same thing with b."},{"Start":"01:10.670 ","End":"01:16.600","Text":"2b plus 1 over 4b minus 2."},{"Start":"01:16.600 ","End":"01:19.340","Text":"Now, when 2 fractions are equal,"},{"Start":"01:19.340 ","End":"01:23.255","Text":"we can cross multiply the diagonals and they will be equal."},{"Start":"01:23.255 ","End":"01:25.205","Text":"This times this will equal this times this."},{"Start":"01:25.205 ","End":"01:33.270","Text":"In other words, 2a plus 1 times 4b minus 2 is going to equal"},{"Start":"01:33.270 ","End":"01:41.970","Text":"4a minus 2 times 2b plus 1."},{"Start":"01:41.970 ","End":"01:46.260","Text":"Let\u0027s multiply out and see what we get."},{"Start":"01:46.260 ","End":"01:50.350","Text":"2a times 4b is 8ab."},{"Start":"01:51.320 ","End":"01:53.715","Text":"Let\u0027s take the a next,"},{"Start":"01:53.715 ","End":"02:00.015","Text":"minus 4a plus 4b minus 2."},{"Start":"02:00.015 ","End":"02:01.350","Text":"On the right-hand side,"},{"Start":"02:01.350 ","End":"02:05.490","Text":"4a times 2b, again 8ab."},{"Start":"02:05.490 ","End":"02:08.610","Text":"The a is next will be plus 4a,"},{"Start":"02:08.610 ","End":"02:12.220","Text":"next the b, minus 4b."},{"Start":"02:12.220 ","End":"02:17.370","Text":"Then the 3 number is minus 2."},{"Start":"02:17.990 ","End":"02:21.030","Text":"Now I see that stuff cancels."},{"Start":"02:21.030 ","End":"02:23.805","Text":"8ab will cancel with 8ab,"},{"Start":"02:23.805 ","End":"02:27.910","Text":"minus 2 with minus 2."},{"Start":"02:28.010 ","End":"02:35.360","Text":"What we\u0027re left with if we bring the a\u0027s over to the left,"},{"Start":"02:35.360 ","End":"02:38.525","Text":"we\u0027ll get minus 8a,"},{"Start":"02:38.525 ","End":"02:40.445","Text":"it\u0027s this with this."},{"Start":"02:40.445 ","End":"02:44.130","Text":"If I bring b\u0027s to the right,"},{"Start":"02:44.130 ","End":"02:46.665","Text":"I get minus 8b."},{"Start":"02:46.665 ","End":"02:53.440","Text":"Now if I divide both side by minus 8,"},{"Start":"02:53.440 ","End":"03:00.010","Text":"then what I\u0027m left with is just a equals b."},{"Start":"03:00.010 ","End":"03:05.360","Text":"In summary, I started out with f of a equals f of b,"},{"Start":"03:05.360 ","End":"03:08.105","Text":"and ended up with a equals b."},{"Start":"03:08.105 ","End":"03:12.800","Text":"The answer to a is yes, it is 1-1."},{"Start":"03:12.800 ","End":"03:15.210","Text":"Onto the next."},{"Start":"03:15.350 ","End":"03:18.930","Text":"Next is b."},{"Start":"03:18.930 ","End":"03:24.075","Text":"We have that f of x is 1 over x cubed."},{"Start":"03:24.075 ","End":"03:26.330","Text":"As usual with 1-1,"},{"Start":"03:26.330 ","End":"03:29.090","Text":"I\u0027m going to start out with f of a equals f of b."},{"Start":"03:29.090 ","End":"03:34.130","Text":"Hopefully, at the end I can get to a equals b."},{"Start":"03:34.130 ","End":"03:35.300","Text":"What does this mean?"},{"Start":"03:35.300 ","End":"03:42.855","Text":"This means that 1 over a cubed is equal to 1 over b cubed."},{"Start":"03:42.855 ","End":"03:49.880","Text":"Once again, I can cross multiply 1 times b cubed equals 1 times a cubed."},{"Start":"03:49.880 ","End":"03:54.925","Text":"In other words, I just get that a cubed is equal to b cubed."},{"Start":"03:54.925 ","End":"03:58.210","Text":"Now, if the cubes of numbers are equal,"},{"Start":"03:58.210 ","End":"03:59.905","Text":"so are the numbers."},{"Start":"03:59.905 ","End":"04:02.035","Text":"This is not trivial."},{"Start":"04:02.035 ","End":"04:04.120","Text":"This conclusion that a equals b,"},{"Start":"04:04.120 ","End":"04:05.350","Text":"because if, for example,"},{"Start":"04:05.350 ","End":"04:08.350","Text":"if it was a squared equals b squared,"},{"Start":"04:08.350 ","End":"04:11.050","Text":"then a could have been plus or minus b."},{"Start":"04:11.050 ","End":"04:14.140","Text":"There would have been a matter of a sign, but with an odd number,"},{"Start":"04:14.140 ","End":"04:15.880","Text":"you\u0027re safe because a plus cubed,"},{"Start":"04:15.880 ","End":"04:18.145","Text":"is plus, and a minus cubed is minus."},{"Start":"04:18.145 ","End":"04:20.620","Text":"We don\u0027t have any of these plus or minus because you just"},{"Start":"04:20.620 ","End":"04:23.435","Text":"take the cube root of both sides and they\u0027re equal."},{"Start":"04:23.435 ","End":"04:26.750","Text":"In short, we started with f of"},{"Start":"04:26.750 ","End":"04:31.600","Text":"a equals f of b and ended up conclusively with a equals b."},{"Start":"04:31.600 ","End":"04:34.120","Text":"In this case also in b,"},{"Start":"04:34.120 ","End":"04:37.405","Text":"the answer is yes, it is 1-1."},{"Start":"04:37.405 ","End":"04:40.890","Text":"Finally, number c,"},{"Start":"04:40.890 ","End":"04:42.840","Text":"f of x equals x squared."},{"Start":"04:42.840 ","End":"04:48.620","Text":"But here we\u0027re given an important condition that x is not negative."},{"Start":"04:48.620 ","End":"04:51.980","Text":"Let\u0027s start with f of a equals f of b,"},{"Start":"04:51.980 ","End":"04:56.285","Text":"which is the way we usually start in proving 1-1."},{"Start":"04:56.285 ","End":"04:58.310","Text":"Let\u0027s see what this means."},{"Start":"04:58.310 ","End":"05:03.980","Text":"This means that a squared is equal to b squared."},{"Start":"05:03.980 ","End":"05:11.570","Text":"Now, we can\u0027t automatically say that a equals b just by taking a square root."},{"Start":"05:11.570 ","End":"05:16.490","Text":"Because theoretically it could have been that a is plus or minus b."},{"Start":"05:16.490 ","End":"05:20.835","Text":"That\u0027s what we would normally say a is plus or minus b."},{"Start":"05:20.835 ","End":"05:25.130","Text":"Because a and b they\u0027re both bigger or equal to 0,"},{"Start":"05:25.130 ","End":"05:28.560","Text":"then it has to be that a equals b."},{"Start":"05:28.960 ","End":"05:35.015","Text":"In summary, we started out with f of a equals f of b,"},{"Start":"05:35.015 ","End":"05:37.775","Text":"and ended up with a equals b."},{"Start":"05:37.775 ","End":"05:39.665","Text":"In this case too,"},{"Start":"05:39.665 ","End":"05:41.659","Text":"the answer is yes,"},{"Start":"05:41.659 ","End":"05:46.050","Text":"it is 1-1. We are done."}],"ID":4839},{"Watched":false,"Name":"Exercise 4 part a","Duration":"1m 43s","ChapterTopicVideoID":4501,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.235","Text":"In this exercise, we have to determine whether the given functions are 1-1 or not."},{"Start":"00:05.235 ","End":"00:07.380","Text":"There are 3 separate exercises: a, b,"},{"Start":"00:07.380 ","End":"00:11.925","Text":"and c. Let\u0027s remember what it means for a function to be 1-1."},{"Start":"00:11.925 ","End":"00:18.315","Text":"It means that if we start with f of a equals f of b,"},{"Start":"00:18.315 ","End":"00:22.845","Text":"this implies that a equals b."},{"Start":"00:22.845 ","End":"00:24.855","Text":"Let\u0027s start with part a."},{"Start":"00:24.855 ","End":"00:31.305","Text":"Applying this method, we get that twice a minus 3 squared"},{"Start":"00:31.305 ","End":"00:38.485","Text":"minus 4 is equal to twice b minus 3 squared minus 4."},{"Start":"00:38.485 ","End":"00:41.180","Text":"Now, let\u0027s do a little bit of algebra here."},{"Start":"00:41.180 ","End":"00:42.440","Text":"Let\u0027s add 4,"},{"Start":"00:42.440 ","End":"00:45.095","Text":"and we can also divide by 2 at the same time,"},{"Start":"00:45.095 ","End":"00:52.730","Text":"so what we get is a minus 3 squared is equal to b minus 3 squared."},{"Start":"00:52.730 ","End":"01:00.890","Text":"Now, the temptation is to say that a minus 3 would equal plus or minus b minus 3."},{"Start":"01:00.890 ","End":"01:03.290","Text":"This is what would normally be true because it\u0027s a square,"},{"Start":"01:03.290 ","End":"01:04.400","Text":"so 2 numbers are equal,"},{"Start":"01:04.400 ","End":"01:06.680","Text":"1 can be plus or minus the other."},{"Start":"01:06.680 ","End":"01:10.130","Text":"However, if you notice we have a restriction on"},{"Start":"01:10.130 ","End":"01:13.970","Text":"the domain of x. X has to be bigger or equal to 3,"},{"Start":"01:13.970 ","End":"01:17.645","Text":"which can also be written as x minus 3,"},{"Start":"01:17.645 ","End":"01:20.000","Text":"bigger or equal to 0."},{"Start":"01:20.000 ","End":"01:23.915","Text":"In this case, both a minus 3 is bigger or equal to 0,"},{"Start":"01:23.915 ","End":"01:26.315","Text":"e minus 3 are bigger or equal to 0,"},{"Start":"01:26.315 ","End":"01:28.940","Text":"so we don\u0027t have the plus or minus option."},{"Start":"01:28.940 ","End":"01:33.140","Text":"We can remove that and just say that a minus 3 equals b minus 3,"},{"Start":"01:33.140 ","End":"01:35.975","Text":"giving us that a equals b."},{"Start":"01:35.975 ","End":"01:39.530","Text":"In other words, the function is 1-1."},{"Start":"01:39.530 ","End":"01:44.340","Text":"I\u0027ll just write the answer yes, and we\u0027re done with part a."}],"ID":4510},{"Watched":false,"Name":"Exercise 4 part b","Duration":"2m 20s","ChapterTopicVideoID":4502,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"Let\u0027s move on to part b."},{"Start":"00:02.880 ","End":"00:06.639","Text":"Once again, beginning with f of a equals f of b,"},{"Start":"00:06.639 ","End":"00:12.560","Text":"we\u0027ll write that by substituting a and then b in this expression and we\u0027ll get that"},{"Start":"00:12.560 ","End":"00:22.245","Text":"a squared minus 4a plus 5 equals b squared minus 4b plus 5."},{"Start":"00:22.245 ","End":"00:27.795","Text":"Let\u0027s subtract the 5 from both sides and get a squared minus 4a,"},{"Start":"00:27.795 ","End":"00:32.275","Text":"is equal to b squared minus 4b."},{"Start":"00:32.275 ","End":"00:35.340","Text":"It\u0027s not quite clear how to proceed from here."},{"Start":"00:35.340 ","End":"00:37.710","Text":"If we didn\u0027t have the a squared or the b squared,"},{"Start":"00:37.710 ","End":"00:42.180","Text":"we\u0027d have minus 4a equals minus 4b and then we could conclude a equals b."},{"Start":"00:42.180 ","End":"00:44.520","Text":"If we just had the a squared and b squared,"},{"Start":"00:44.520 ","End":"00:47.525","Text":"we could get a is plus or minus b and take it from there."},{"Start":"00:47.525 ","End":"00:49.190","Text":"But what do we do here?"},{"Start":"00:49.190 ","End":"00:51.920","Text":"I want to remind you of a technique called completing"},{"Start":"00:51.920 ","End":"00:56.585","Text":"the square and you can refer to the appropriate theoretical chapter on that."},{"Start":"00:56.585 ","End":"01:00.724","Text":"The idea is to make both sides perfect squares by adding something."},{"Start":"01:00.724 ","End":"01:03.650","Text":"In this case, we\u0027ll add 4 to"},{"Start":"01:03.650 ","End":"01:09.610","Text":"both sides and this should be a familiar expression because this is a perfect square,"},{"Start":"01:09.610 ","End":"01:13.570","Text":"this is exactly a minus 2 squared,"},{"Start":"01:13.570 ","End":"01:17.585","Text":"and here we have b minus 2 squared."},{"Start":"01:17.585 ","End":"01:24.925","Text":"Now, notice that the domain is restricted to x less than or equal to 2,"},{"Start":"01:24.925 ","End":"01:31.115","Text":"which means that x minus 2 is less than or equal to 0."},{"Start":"01:31.115 ","End":"01:34.340","Text":"It\u0027s not positive, which means that a minus 2 is less"},{"Start":"01:34.340 ","End":"01:38.350","Text":"than or equal to 0 and b minus 2 is less than or equal to 0."},{"Start":"01:38.350 ","End":"01:40.805","Text":"If we didn\u0027t have this restriction,"},{"Start":"01:40.805 ","End":"01:46.850","Text":"we would be writing something like a minus 2 is plus or minus b minus 2."},{"Start":"01:46.850 ","End":"01:48.620","Text":"Because if squares are equal,"},{"Start":"01:48.620 ","End":"01:50.675","Text":"1 can be plus or minus the other."},{"Start":"01:50.675 ","End":"01:54.080","Text":"But because both these 2 are negative or 0,"},{"Start":"01:54.080 ","End":"01:59.450","Text":"we can\u0027t have the minus so I\u0027m just going to remove this part here."},{"Start":"01:59.450 ","End":"02:02.180","Text":"a minus 2 has to equal b minus 2,"},{"Start":"02:02.180 ","End":"02:07.340","Text":"which means that a equals b and this is what we had to"},{"Start":"02:07.340 ","End":"02:13.070","Text":"show for it being 1 to 1. f of a equals f of b and a equals b,"},{"Start":"02:13.070 ","End":"02:15.965","Text":"that means that in part b,"},{"Start":"02:15.965 ","End":"02:18.425","Text":"the answer is also yes,"},{"Start":"02:18.425 ","End":"02:20.520","Text":"it is 1 to 1."}],"ID":4511},{"Watched":false,"Name":"Exercise 4 part c","Duration":"1m 11s","ChapterTopicVideoID":4503,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"Moving on to part c. Using the same technique,"},{"Start":"00:03.825 ","End":"00:08.625","Text":"we start with f of a equals f of b and see if we get to a equals b."},{"Start":"00:08.625 ","End":"00:11.535","Text":"So putting x 1s as a and 1s as b,"},{"Start":"00:11.535 ","End":"00:16.170","Text":"we get a squared over a squared plus"},{"Start":"00:16.170 ","End":"00:22.110","Text":"1 is equal to b squared over b squared plus 1."},{"Start":"00:22.110 ","End":"00:27.990","Text":"Cross-multiplying, we get a squared, b squared plus a"},{"Start":"00:27.990 ","End":"00:34.380","Text":"squared is equal to a squared b squared plus b squared."},{"Start":"00:34.380 ","End":"00:38.459","Text":"We can cancel the common term a squared b squared from both sides."},{"Start":"00:38.459 ","End":"00:42.295","Text":"What we\u0027re left with is that a squared equals b squared."},{"Start":"00:42.295 ","End":"00:45.395","Text":"Now, normally, when a squared equals b squared,"},{"Start":"00:45.395 ","End":"00:48.830","Text":"we would say that a is plus or minus b."},{"Start":"00:48.830 ","End":"00:51.110","Text":"However, in our case,"},{"Start":"00:51.110 ","End":"00:52.685","Text":"the domain is restricted."},{"Start":"00:52.685 ","End":"00:54.350","Text":"X is bigger or equal to 0,"},{"Start":"00:54.350 ","End":"00:57.685","Text":"which means that both a and b are bigger or equal to 0."},{"Start":"00:57.685 ","End":"01:05.450","Text":"This is unnecessary and we have to get that a equals B because they\u0027re both non-negative."},{"Start":"01:05.450 ","End":"01:08.865","Text":"That means that this function is 1 to 1."},{"Start":"01:08.865 ","End":"01:11.910","Text":"We\u0027re done with the whole exercise."}],"ID":4512},{"Watched":false,"Name":"Exercise 5 part a","Duration":"59s","ChapterTopicVideoID":4504,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this exercise, we have to determine whether each of"},{"Start":"00:02.640 ","End":"00:06.120","Text":"these 3 functions is one-to-one or not."},{"Start":"00:06.120 ","End":"00:10.170","Text":"Reminder, a function is called one-to-one,"},{"Start":"00:10.170 ","End":"00:13.965","Text":"if whenever f of a equals f of b,"},{"Start":"00:13.965 ","End":"00:18.735","Text":"we can conclude that necessarily a equals b."},{"Start":"00:18.735 ","End":"00:20.895","Text":"Begin with part a."},{"Start":"00:20.895 ","End":"00:24.630","Text":"We start off with f of a equals f of b."},{"Start":"00:24.630 ","End":"00:25.770","Text":"Putting x once as a,"},{"Start":"00:25.770 ","End":"00:26.895","Text":"once as b,"},{"Start":"00:26.895 ","End":"00:34.620","Text":"we get 4 natural log of a equals 4 natural log of b."},{"Start":"00:34.620 ","End":"00:38.040","Text":"We can cancel the 4 and I will divide both sides by"},{"Start":"00:38.040 ","End":"00:42.380","Text":"4 and we get the natural log of a equals natural log of b."},{"Start":"00:42.380 ","End":"00:45.230","Text":"Now, if you remember from the lesson on natural log,"},{"Start":"00:45.230 ","End":"00:48.575","Text":"that natural log is a one-to-one function,"},{"Start":"00:48.575 ","End":"00:55.970","Text":"which means that a equals b and this means that our function is one-to-one."},{"Start":"00:55.970 ","End":"00:59.550","Text":"The answer is yes. That\u0027s part a."}],"ID":4513},{"Watched":false,"Name":"Exercise 5 part b","Duration":"55s","ChapterTopicVideoID":4505,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.115","Text":"Moving on to Part b."},{"Start":"00:02.115 ","End":"00:05.820","Text":"Once again, f of a equals f of b is our starting point,"},{"Start":"00:05.820 ","End":"00:11.700","Text":"and we get 2 plus 3 natural log of a minus"},{"Start":"00:11.700 ","End":"00:17.955","Text":"1 is equal to 2 plus natural log of b minus 1."},{"Start":"00:17.955 ","End":"00:21.295","Text":"We can subtract the 2 from both sides,"},{"Start":"00:21.295 ","End":"00:24.575","Text":"and I forgot the 3 here. Excuse me."},{"Start":"00:24.575 ","End":"00:30.470","Text":"We can also divide both sides by 3 so we get that natural log of"},{"Start":"00:30.470 ","End":"00:36.625","Text":"a minus 1 is equal to natural log of b minus 1."},{"Start":"00:36.625 ","End":"00:43.290","Text":"As before, the natural log is 1 to 1 which means that a minus 1 equals"},{"Start":"00:43.290 ","End":"00:50.115","Text":"b minus 1 from which clearly a equals b which is what we needed to show,"},{"Start":"00:50.115 ","End":"00:55.660","Text":"so that gives the answer at b also as yes."}],"ID":4514},{"Watched":false,"Name":"Exercise 5 part c","Duration":"48s","ChapterTopicVideoID":4506,"CourseChapterTopicPlaylistID":1184,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.720","Text":"Moving on to part C. Using the same technique we get 1 plus"},{"Start":"00:06.720 ","End":"00:15.015","Text":"2e to the 2a equals 1 plus twice e to the power of 2b."},{"Start":"00:15.015 ","End":"00:17.370","Text":"We can cancel the 1,"},{"Start":"00:17.370 ","End":"00:19.855","Text":"and we can also divide by 2."},{"Start":"00:19.855 ","End":"00:25.650","Text":"We get e to the power of 2a is equal to e to the power of 2b."},{"Start":"00:25.650 ","End":"00:29.415","Text":"Now if you remember the lesson on the exponential function,"},{"Start":"00:29.415 ","End":"00:33.165","Text":"e to the power of this function is also 1-to-1,"},{"Start":"00:33.165 ","End":"00:37.390","Text":"which means that 2a has to equal 2b."},{"Start":"00:37.390 ","End":"00:40.565","Text":"In other words, a equals b,"},{"Start":"00:40.565 ","End":"00:42.065","Text":"which is what we wanted."},{"Start":"00:42.065 ","End":"00:45.320","Text":"In conclusion, part C is 1-to-1,"},{"Start":"00:45.320 ","End":"00:49.330","Text":"and in fact, all 3 parts are 1-to-1. We\u0027re done."}],"ID":4515}],"Thumbnail":null,"ID":1184},{"Name":"The Inverse of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Inverse of a Function","Duration":"12m 19s","ChapterTopicVideoID":1228,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this clip I\u0027ll be introducing the concept of an inverse function."},{"Start":"00:03.780 ","End":"00:05.700","Text":"I\u0027ll begin with the example."},{"Start":"00:05.700 ","End":"00:12.255","Text":"Let\u0027s take the function f(x) which is equal to 4x plus 1."},{"Start":"00:12.255 ","End":"00:14.415","Text":"This function, or any function,"},{"Start":"00:14.415 ","End":"00:17.205","Text":"you give it a number and it gives you back a number."},{"Start":"00:17.205 ","End":"00:21.380","Text":"For example if I feed this function the number 0,"},{"Start":"00:21.380 ","End":"00:25.020","Text":"it spits out 4 times 0 plus 1 is 1."},{"Start":"00:25.020 ","End":"00:27.570","Text":"If I give it the number 1,"},{"Start":"00:27.570 ","End":"00:32.025","Text":"it gives me back 4 times 1 plus 1 gives me back 5."},{"Start":"00:32.025 ","End":"00:34.760","Text":"If I give it minus 2,"},{"Start":"00:34.760 ","End":"00:40.820","Text":"I get back minus 2 times 4 is minus 8 plus 1 is minus 7."},{"Start":"00:40.820 ","End":"00:42.875","Text":"I can represent it this way."},{"Start":"00:42.875 ","End":"00:48.530","Text":"The function f takes the number 0 and gives me back 1 and I can write it like this."},{"Start":"00:48.530 ","End":"00:52.711","Text":"Similarly, it take 1 and it gives back 5."},{"Start":"00:52.711 ","End":"00:54.445","Text":"Feed it minus 2,"},{"Start":"00:54.445 ","End":"00:57.073","Text":"and it spits out minus 7."},{"Start":"00:57.073 ","End":"01:00.489","Text":"The inverse function of f will do exactly the opposite."},{"Start":"01:00.489 ","End":"01:04.415","Text":"It will take the number 1 and give me back a 0."},{"Start":"01:04.415 ","End":"01:05.960","Text":"I\u0027ll feed it 5,"},{"Start":"01:05.960 ","End":"01:07.525","Text":"it\u0027ll spit out 1."},{"Start":"01:07.525 ","End":"01:09.390","Text":"I\u0027ll give it minus 7,"},{"Start":"01:09.390 ","End":"01:11.335","Text":"it\u0027ll give back minus 2."},{"Start":"01:11.335 ","End":"01:13.685","Text":"The arrows are exactly the opposite."},{"Start":"01:13.685 ","End":"01:17.420","Text":"There\u0027s a special notation for the inverse function and"},{"Start":"01:17.420 ","End":"01:22.880","Text":"the inverse function is written as f and the minus 1 at the top here,"},{"Start":"01:22.880 ","End":"01:26.570","Text":"and the minus 1 doesn\u0027t mean like 1 over or something"},{"Start":"01:26.570 ","End":"01:30.470","Text":"like that it\u0027s just a symbol used to represent the inverse function."},{"Start":"01:30.470 ","End":"01:33.920","Text":"Now, I\u0027m going to tell you what the inverse function is."},{"Start":"01:33.920 ","End":"01:35.780","Text":"Never mind for the moment how I got it,"},{"Start":"01:35.780 ","End":"01:36.980","Text":"that we\u0027ll learn later,"},{"Start":"01:36.980 ","End":"01:42.795","Text":"and the inverse function is going to be x minus 1 over 4,"},{"Start":"01:42.795 ","End":"01:44.550","Text":"and I\u0027m going to highlight this."},{"Start":"01:44.550 ","End":"01:47.635","Text":"Now, I can claim that this is the inverse function,"},{"Start":"01:47.635 ","End":"01:50.360","Text":"but I have to demonstrate to you that it really is."},{"Start":"01:50.360 ","End":"01:56.330","Text":"Let\u0027s see what happens if we substitute these 3 numbers and we get back these 3."},{"Start":"01:56.330 ","End":"01:59.285","Text":"So f minus 1 of 1,"},{"Start":"01:59.285 ","End":"02:00.755","Text":"that\u0027s the 1 from here,"},{"Start":"02:00.755 ","End":"02:08.715","Text":"is equal to 1 minus 1 over 4 and this is indeed 0 as expected."},{"Start":"02:08.715 ","End":"02:12.240","Text":"For the next one if I put x equals 5,"},{"Start":"02:12.240 ","End":"02:14.640","Text":"I get 5 minus 1 over 4,"},{"Start":"02:14.640 ","End":"02:17.370","Text":"which is 4 over 4, which is 1."},{"Start":"02:17.370 ","End":"02:25.430","Text":"So f minus 1 of 5 is indeed equal to 1 and f minus 1 of minus"},{"Start":"02:25.430 ","End":"02:30.020","Text":"7 is minus 7 minus 1 which is minus"},{"Start":"02:30.020 ","End":"02:35.690","Text":"8 minus 8 over 4 is minus 2 as expected."},{"Start":"02:35.690 ","End":"02:39.155","Text":"As above I\u0027ll represent this with arrows as follows,"},{"Start":"02:39.155 ","End":"02:43.130","Text":"that 1 goes to 0,"},{"Start":"02:43.130 ","End":"02:50.220","Text":"5 goes to 1 and minus 7 goes to minus 2,"},{"Start":"02:50.220 ","End":"02:53.600","Text":"all this work is done by the function f minus 1,"},{"Start":"02:53.600 ","End":"02:56.960","Text":"the inverse function of f. In general,"},{"Start":"02:56.960 ","End":"03:01.300","Text":"I can say that if f(a) is equal to b."},{"Start":"03:01.300 ","End":"03:05.140","Text":"In other words if f takes a and gives back b,"},{"Start":"03:05.140 ","End":"03:14.445","Text":"then the inverse function f minus 1 will take b as input and give a as its output."},{"Start":"03:14.445 ","End":"03:17.235","Text":"This is a function,"},{"Start":"03:17.235 ","End":"03:20.141","Text":"and this is the inverse function."},{"Start":"03:20.141 ","End":"03:24.075","Text":"Let me give you a little sketch so you can see what happens graphically."},{"Start":"03:24.075 ","End":"03:28.820","Text":"Here the 2 functions this 1 is y equals f(x) and I took some point (a,"},{"Start":"03:28.820 ","End":"03:32.075","Text":"b) on the graph of f. If (a, b) is here,"},{"Start":"03:32.075 ","End":"03:33.880","Text":"then I\u0027m guaranteed that (b,"},{"Start":"03:33.880 ","End":"03:38.460","Text":"a) will be on the graph of the inverse function which is called,"},{"Start":"03:38.460 ","End":"03:40.980","Text":"as you remember, f to the minus 1."},{"Start":"03:40.980 ","End":"03:45.035","Text":"Again, I want to stress this minus 1 is just a symbol,"},{"Start":"03:45.035 ","End":"03:47.930","Text":"it doesn\u0027t mean 1 over f or anything like that."},{"Start":"03:47.930 ","End":"03:51.630","Text":"Now, let\u0027s get back to something that I owe you,"},{"Start":"03:51.630 ","End":"03:55.119","Text":"you remember that at the beginning I took a function,"},{"Start":"03:55.119 ","End":"03:57.373","Text":"this one, and out of the hat,"},{"Start":"03:57.373 ","End":"03:59.855","Text":"I produced the inverse function."},{"Start":"03:59.855 ","End":"04:03.140","Text":"I\u0027m going to show you how we can take a function such as"},{"Start":"04:03.140 ","End":"04:07.930","Text":"this and compute its inverse and I\u0027m going to use this as the example,"},{"Start":"04:07.930 ","End":"04:11.740","Text":"but meanwhile I would like to hide this as if we don\u0027t know about it."},{"Start":"04:11.740 ","End":"04:13.780","Text":"So let\u0027s do it in steps."},{"Start":"04:13.780 ","End":"04:16.495","Text":"The steps will work for pretty much all cases."},{"Start":"04:16.495 ","End":"04:22.420","Text":"The very first step, Step 1 is just to replace the f notation with the y notation."},{"Start":"04:22.420 ","End":"04:26.665","Text":"We\u0027ll write y equals 4x plus 1."},{"Start":"04:26.665 ","End":"04:29.110","Text":"You may be lucky and be given it in the form of"},{"Start":"04:29.110 ","End":"04:31.630","Text":"y equals and that will save you a step and"},{"Start":"04:31.630 ","End":"04:36.709","Text":"also a step at the end where we convert the y back to functional notation,"},{"Start":"04:36.709 ","End":"04:38.110","Text":"that\u0027s the first thing."},{"Start":"04:38.110 ","End":"04:44.095","Text":"The second thing to do is to replace x with y and y with x just interchange them."},{"Start":"04:44.095 ","End":"04:48.660","Text":"This gives me x equals 4y plus 1."},{"Start":"04:48.660 ","End":"04:50.160","Text":"You might ask what\u0027s the logic of this?"},{"Start":"04:50.160 ","End":"04:52.020","Text":"We\u0027re not going to go into great detail,"},{"Start":"04:52.020 ","End":"04:54.430","Text":"but remember on the graph when we saw that (a,"},{"Start":"04:54.430 ","End":"04:56.300","Text":"b) was on the function f and (b,"},{"Start":"04:56.300 ","End":"04:59.200","Text":"a) was in the function f to the minus 1 the inverse,"},{"Start":"04:59.200 ","End":"05:02.315","Text":"well, the same thing here positions of x and y get interchanged."},{"Start":"05:02.315 ","End":"05:06.020","Text":"The third step, this is sometimes a bit complicated algebraically,"},{"Start":"05:06.020 ","End":"05:11.329","Text":"here it\u0027s not, is to isolate y in terms of x by using algebra."},{"Start":"05:11.329 ","End":"05:12.950","Text":"I leave the 4y here,"},{"Start":"05:12.950 ","End":"05:14.285","Text":"but write it on the left,"},{"Start":"05:14.285 ","End":"05:16.610","Text":"and then the x minus 1,"},{"Start":"05:16.610 ","End":"05:25.340","Text":"we\u0027ll put on the right and then divide by 4 so I get y equals x minus 1 over 4."},{"Start":"05:25.340 ","End":"05:29.480","Text":"Finally, now just write the inverse in functional notation."},{"Start":"05:29.480 ","End":"05:34.700","Text":"This gives us f minus 1 of x is equal to x minus 1 over 4 and,"},{"Start":"05:34.700 ","End":"05:37.370","Text":"if you remember, this is exactly what we"},{"Start":"05:37.370 ","End":"05:40.540","Text":"had for the inverse function that I produced out of that."},{"Start":"05:40.540 ","End":"05:42.635","Text":"This is the general technique."},{"Start":"05:42.635 ","End":"05:47.040","Text":"Let\u0027s take another example and we\u0027ll take the function f(x) is"},{"Start":"05:47.040 ","End":"05:53.170","Text":"equal to x plus 1 over x plus 2."},{"Start":"05:53.170 ","End":"05:58.280","Text":"My task is to find out what is the inverse function of f or f to the minus 1."},{"Start":"05:58.280 ","End":"06:00.695","Text":"We\u0027ll follow the same steps as here."},{"Start":"06:00.695 ","End":"06:09.941","Text":"Step 1 is that we\u0027ll write y equals x plus 1 over x plus 2."},{"Start":"06:09.941 ","End":"06:14.940","Text":"Step 2, swap between x and y so we get"},{"Start":"06:14.940 ","End":"06:21.780","Text":"x equals y plus 1 over y plus 2."},{"Start":"06:21.780 ","End":"06:28.580","Text":"Step 3, do some algebra to isolate y on the left and everything else on the right."},{"Start":"06:28.580 ","End":"06:32.905","Text":"I\u0027m going to multiply both sides by y plus 2."},{"Start":"06:32.905 ","End":"06:34.890","Text":"Y plus 2 will go here,"},{"Start":"06:34.890 ","End":"06:38.895","Text":"it\u0027ll be xy plus 2x,"},{"Start":"06:38.895 ","End":"06:42.660","Text":"and this will equal just y plus 1,"},{"Start":"06:42.660 ","End":"06:47.475","Text":"then I want all the y\u0027s on the left and everything else on the right,"},{"Start":"06:47.475 ","End":"06:52.736","Text":"so xy minus y equals"},{"Start":"06:52.736 ","End":"06:59.480","Text":"1 minus 2x and then I\u0027d like to take y outside the brackets,"},{"Start":"06:59.480 ","End":"07:02.315","Text":"if I do it\u0027ll be y times x minus 1."},{"Start":"07:02.315 ","End":"07:07.565","Text":"Straight away, I want to put the x minus 1 in the denominator of the other side,"},{"Start":"07:07.565 ","End":"07:10.605","Text":"and that will leave me just with y."},{"Start":"07:10.605 ","End":"07:12.234","Text":"Finally, Step 4,"},{"Start":"07:12.234 ","End":"07:16.115","Text":"inverse function of f(x) is equal to,"},{"Start":"07:16.115 ","End":"07:17.360","Text":"I just copy that,"},{"Start":"07:17.360 ","End":"07:24.215","Text":"1 minus 2x over x minus 1, there we are."},{"Start":"07:24.215 ","End":"07:26.810","Text":"But I\u0027d like to be still stricter."},{"Start":"07:26.810 ","End":"07:29.795","Text":"I mean, I\u0027ve given you some kind of recipe,"},{"Start":"07:29.795 ","End":"07:32.630","Text":"but how do you know this really is the inverse function?"},{"Start":"07:32.630 ","End":"07:36.425","Text":"Let\u0027s check it out some more and see that it does do the job."},{"Start":"07:36.425 ","End":"07:38.704","Text":"Now first I\u0027ll test it on an actual number."},{"Start":"07:38.704 ","End":"07:42.155","Text":"Let\u0027s say I let x equals 1 in the regular function,"},{"Start":"07:42.155 ","End":"07:48.990","Text":"f(1) is equal to 1 plus 1 over 1 plus 2 is 2/3."},{"Start":"07:48.990 ","End":"07:52.050","Text":"So f takes in 1 and spits out 2/3."},{"Start":"07:52.050 ","End":"07:54.740","Text":"I want to see that f minus 1 does the opposite job,"},{"Start":"07:54.740 ","End":"07:57.440","Text":"if I give it 2/3 it\u0027ll give me back 1."},{"Start":"07:57.440 ","End":"08:03.795","Text":"Let\u0027s see f minus 1 of 2/3 is equal to,"},{"Start":"08:03.795 ","End":"08:05.310","Text":"bit messy not too bad."},{"Start":"08:05.310 ","End":"08:12.030","Text":"1 minus twice 2/3 over 2/3 minus"},{"Start":"08:12.030 ","End":"08:21.705","Text":"1 and this equals minus 1/3 over minus 1/3 and the final answer is 1,"},{"Start":"08:21.705 ","End":"08:22.950","Text":"which is what we wanted,"},{"Start":"08:22.950 ","End":"08:25.620","Text":"so it doesn\u0027t go in the opposite direction so it looks good,"},{"Start":"08:25.620 ","End":"08:28.350","Text":"but this is still not a general verification."},{"Start":"08:28.350 ","End":"08:32.465","Text":"What I\u0027m going to do is generalize by putting, instead of 1,"},{"Start":"08:32.465 ","End":"08:35.720","Text":"I\u0027ll put x here and then whatever answer I get,"},{"Start":"08:35.720 ","End":"08:38.990","Text":"I\u0027ll put back in here and see if I get back to x."},{"Start":"08:38.990 ","End":"08:42.770","Text":"That\u0027s my strategy. I also want to point out something else,"},{"Start":"08:42.770 ","End":"08:47.225","Text":"that if this function is the inverse of this function,"},{"Start":"08:47.225 ","End":"08:48.920","Text":"the reverse is also true."},{"Start":"08:48.920 ","End":"08:51.530","Text":"That if I start with this is my original function,"},{"Start":"08:51.530 ","End":"08:53.283","Text":"this will be my inverse,"},{"Start":"08:53.283 ","End":"08:57.395","Text":"the relation of being inverse is reciprocal, it\u0027s symmetric."},{"Start":"08:57.395 ","End":"09:00.350","Text":"1 is the inverse of the other the others the inverse of the 1."},{"Start":"09:00.350 ","End":"09:01.970","Text":"I could also have done it the other way."},{"Start":"09:01.970 ","End":"09:06.005","Text":"If I started off with 2/3 and put it in this function and get out 1,"},{"Start":"09:06.005 ","End":"09:07.805","Text":"if I put 1 in here,"},{"Start":"09:07.805 ","End":"09:09.295","Text":"I get out 2/3."},{"Start":"09:09.295 ","End":"09:11.929","Text":"They each stand each other back to where they came from."},{"Start":"09:11.929 ","End":"09:14.150","Text":"So that\u0027s the point to bear in mind that if there is"},{"Start":"09:14.150 ","End":"09:16.925","Text":"a symmetry here that each is the inverse of the other."},{"Start":"09:16.925 ","End":"09:22.849","Text":"Now I\u0027m going to write down a mathematical proposition which is actually a definition,"},{"Start":"09:22.849 ","End":"09:24.380","Text":"depending on how you look at it,"},{"Start":"09:24.380 ","End":"09:29.420","Text":"f(f) to the minus 1 of x is equal to"},{"Start":"09:29.420 ","End":"09:37.440","Text":"x and also f minus 1 of f(x) is also equal to x."},{"Start":"09:37.440 ","End":"09:39.375","Text":"Now we first of all did it this way."},{"Start":"09:39.375 ","End":"09:43.170","Text":"We put a number into f and what it spit out,"},{"Start":"09:43.170 ","End":"09:46.595","Text":"we put into f minus 1 and we got back to our original x."},{"Start":"09:46.595 ","End":"09:48.110","Text":"But it would have worked the other way."},{"Start":"09:48.110 ","End":"09:51.590","Text":"If we start with the inverse function and then apply the original function,"},{"Start":"09:51.590 ","End":"09:53.750","Text":"you still get back to where you started from."},{"Start":"09:53.750 ","End":"09:56.000","Text":"This could be considered a proposition but"},{"Start":"09:56.000 ","End":"09:58.505","Text":"actually in the books it\u0027s used as a definition."},{"Start":"09:58.505 ","End":"10:01.700","Text":"We say that f1 is the inverse of f and"},{"Start":"10:01.700 ","End":"10:05.060","Text":"has the right to be called the inverse if it satisfies these 2,"},{"Start":"10:05.060 ","End":"10:08.285","Text":"so either way you can look at it as a proposition or a definition."},{"Start":"10:08.285 ","End":"10:10.076","Text":"We use the informal definition."},{"Start":"10:10.076 ","End":"10:13.930","Text":"It\u0027s the one that takes you back to where you started in both directions."},{"Start":"10:13.930 ","End":"10:21.360","Text":"Let\u0027s show that it\u0027s true in our case to remind you we had the f(x) was equal to x"},{"Start":"10:21.360 ","End":"10:28.955","Text":"plus 1 over x plus 2 and we had the inverse of f,"},{"Start":"10:28.955 ","End":"10:33.555","Text":"1 minus 2x over x minus 1."},{"Start":"10:33.555 ","End":"10:35.130","Text":"I\u0027m not going to do both of these,"},{"Start":"10:35.130 ","End":"10:39.770","Text":"I\u0027m going to do the first one and leave the second one for you as an exercise."},{"Start":"10:39.770 ","End":"10:41.750","Text":"How do I prove inequality?"},{"Start":"10:41.750 ","End":"10:45.170","Text":"One way is to start from the left-hand side and end up in the right-hand side,"},{"Start":"10:45.170 ","End":"10:46.550","Text":"and this is what I\u0027m going to do."},{"Start":"10:46.550 ","End":"10:55.305","Text":"This one is equal to f minus 1 of x is 1 minus 2x over x minus 1,"},{"Start":"10:55.305 ","End":"10:57.725","Text":"which means that in the function f,"},{"Start":"10:57.725 ","End":"10:59.120","Text":"everywhere I see x,"},{"Start":"10:59.120 ","End":"11:02.030","Text":"I now put this new thing here, this fraction."},{"Start":"11:02.030 ","End":"11:11.160","Text":"This is equal to, now I\u0027m looking here x is 1 minus 2x over x minus 1 plus 1 and on"},{"Start":"11:11.160 ","End":"11:20.530","Text":"the denominator we have x plus 2 but we replace x by 1 minus 2x over x minus 1 plus 2."},{"Start":"11:20.530 ","End":"11:23.270","Text":"This is what happens when I substitute this into f,"},{"Start":"11:23.270 ","End":"11:25.535","Text":"I just replace x with this."},{"Start":"11:25.535 ","End":"11:27.035","Text":"Okay, little bit of work here,"},{"Start":"11:27.035 ","End":"11:30.563","Text":"multiply top and bottom by x minus 1,"},{"Start":"11:30.563 ","End":"11:32.550","Text":"that will be the common factor."},{"Start":"11:32.550 ","End":"11:36.280","Text":"So on the top, I\u0027ll get just the 1 minus 2x without the x"},{"Start":"11:36.280 ","End":"11:40.210","Text":"minus 1 and 1 times x minus 1 is x minus 1,"},{"Start":"11:40.210 ","End":"11:41.645","Text":"I don\u0027t need brackets."},{"Start":"11:41.645 ","End":"11:48.990","Text":"On the bottom if you multiply by x minus 1 I get 1 minus 2x plus twice x minus"},{"Start":"11:48.990 ","End":"11:53.948","Text":"1 and this equals 1 minus 2x plus x minus 1 the"},{"Start":"11:53.948 ","End":"12:00.090","Text":"1 minus 1 cancels and the minus 2x plus x gives us minus x."},{"Start":"12:00.090 ","End":"12:07.890","Text":"On the bottom I have minus 2x plus 2x is equal to 0 and"},{"Start":"12:07.890 ","End":"12:16.095","Text":"then I have 1 minus 2 which is minus 1 and this equals x which is what I wanted,"},{"Start":"12:16.095 ","End":"12:20.020","Text":"so it works okay in this direction."}],"ID":1228},{"Watched":false,"Name":"The Inverse of a Function(Continued)","Duration":"7m 39s","ChapterTopicVideoID":8246,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"This clip is a supplementary clip to the one on inverse functions."},{"Start":"00:04.680 ","End":"00:06.870","Text":"In the first clip, for the sake of simplicity,"},{"Start":"00:06.870 ","End":"00:11.055","Text":"I didn\u0027t get into some of the fine points that could get us into trouble."},{"Start":"00:11.055 ","End":"00:15.975","Text":"Here I\u0027ll try to correct that and bring some points that you have to watch out for,"},{"Start":"00:15.975 ","End":"00:17.190","Text":"and how to fix them."},{"Start":"00:17.190 ","End":"00:19.350","Text":"Let\u0027s start with an example."},{"Start":"00:19.350 ","End":"00:22.620","Text":"The function f(x) is equal to x^2."},{"Start":"00:22.620 ","End":"00:28.080","Text":"We try in our usual way with the steps to find the inverse function."},{"Start":"00:28.080 ","End":"00:32.670","Text":"We write it in the y notation, y equals x^2."},{"Start":"00:32.670 ","End":"00:35.865","Text":"Then the next step we switch x and y."},{"Start":"00:35.865 ","End":"00:39.260","Text":"We get x equals y^2."},{"Start":"00:39.260 ","End":"00:42.575","Text":"From this point, we try to isolate y,"},{"Start":"00:42.575 ","End":"00:48.820","Text":"maybe write y^2 equals x."},{"Start":"00:48.820 ","End":"00:51.215","Text":"We\u0027ve gotten into a bit of trouble here."},{"Start":"00:51.215 ","End":"00:52.850","Text":"But y equal to,"},{"Start":"00:52.850 ","End":"00:54.545","Text":"if x is positive,"},{"Start":"00:54.545 ","End":"00:58.490","Text":"then y could be plus or minus the square root of x."},{"Start":"00:58.490 ","End":"01:00.080","Text":"Say if x was 4,"},{"Start":"01:00.080 ","End":"01:04.115","Text":"I wouldn\u0027t know if y is 2 or minus 2 because they both fit."},{"Start":"01:04.115 ","End":"01:06.380","Text":"Here\u0027s where I have the problem."},{"Start":"01:06.380 ","End":"01:08.210","Text":"How do I solve this problem?"},{"Start":"01:08.210 ","End":"01:13.115","Text":"In practice, the exercise is usually given with an additional constraint."},{"Start":"01:13.115 ","End":"01:16.100","Text":"For example, the original exercise might\u0027ve been,"},{"Start":"01:16.100 ","End":"01:18.260","Text":"find the inverse of f(x),"},{"Start":"01:18.260 ","End":"01:19.685","Text":"which is equal to x^2,"},{"Start":"01:19.685 ","End":"01:24.530","Text":"given that x is bigger or equal to 0."},{"Start":"01:24.530 ","End":"01:27.485","Text":"Now, if I was given that x is bigger or equal to 0,"},{"Start":"01:27.485 ","End":"01:29.435","Text":"then when I switched x and y,"},{"Start":"01:29.435 ","End":"01:33.890","Text":"I would have now the condition that y is bigger or equal to 0."},{"Start":"01:33.890 ","End":"01:36.514","Text":"Then when I got to this place,"},{"Start":"01:36.514 ","End":"01:43.080","Text":"I would be able to say y is bigger or equal to 0 so I can write what the answer is,"},{"Start":"01:43.080 ","End":"01:46.150","Text":"y equals the square root of x, and there\u0027s no problem."},{"Start":"01:46.150 ","End":"01:48.625","Text":"This is non-negative. Just by the way,"},{"Start":"01:48.625 ","End":"01:50.725","Text":"how do you know that x is not negative?"},{"Start":"01:50.725 ","End":"01:51.790","Text":"Well, if you look at this,"},{"Start":"01:51.790 ","End":"01:54.280","Text":"x is y^2, so it\u0027s certainly not negative."},{"Start":"01:54.280 ","End":"01:55.510","Text":"Then here at the end,"},{"Start":"01:55.510 ","End":"02:03.325","Text":"we would then say f to the minus 1 of x. Inverse function of f is the square root of x."},{"Start":"02:03.325 ","End":"02:08.028","Text":"But the original question did not have this extra restriction,"},{"Start":"02:08.028 ","End":"02:09.400","Text":"and something did go wrong."},{"Start":"02:09.400 ","End":"02:11.755","Text":"Let\u0027s see what actually went wrong."},{"Start":"02:11.755 ","End":"02:14.440","Text":"The thing about f is that it has"},{"Start":"02:14.440 ","End":"02:18.235","Text":"the property that it can take 2 different values to the same value."},{"Start":"02:18.235 ","End":"02:21.685","Text":"I\u0027ll show you f of 2 is 2^2 is 4."},{"Start":"02:21.685 ","End":"02:24.095","Text":"If I look at f of minus 2,"},{"Start":"02:24.095 ","End":"02:27.340","Text":"that\u0027s minus 2^2 is also equal to 4."},{"Start":"02:27.340 ","End":"02:29.750","Text":"This property of f that it can take"},{"Start":"02:29.750 ","End":"02:34.080","Text":"2 different values for the same value like 2 and minus 2 both go to 4,"},{"Start":"02:34.080 ","End":"02:36.680","Text":"this is the essence of the problem."},{"Start":"02:36.680 ","End":"02:38.810","Text":"Let me spell it out a bit more."},{"Start":"02:38.810 ","End":"02:40.340","Text":"If f has an inverse,"},{"Start":"02:40.340 ","End":"02:41.540","Text":"f to the minus 1,"},{"Start":"02:41.540 ","End":"02:43.070","Text":"it would have to do the opposite."},{"Start":"02:43.070 ","End":"02:46.745","Text":"It would have to take 4 back to 2 and it would have to"},{"Start":"02:46.745 ","End":"02:50.900","Text":"take 4 back to minus 2 or if you like,"},{"Start":"02:50.900 ","End":"02:52.220","Text":"in the functional notation,"},{"Start":"02:52.220 ","End":"02:53.660","Text":"we would write it like this."},{"Start":"02:53.660 ","End":"02:57.303","Text":"But either way, this contradicts the very definition of a function"},{"Start":"02:57.303 ","End":"03:00.950","Text":"as a function takes a value to exactly 1 value,"},{"Start":"03:00.950 ","End":"03:03.590","Text":"4 can\u0027t go both to 2 and to minus 2."},{"Start":"03:03.590 ","End":"03:05.420","Text":"So this would not be a function."},{"Start":"03:05.420 ","End":"03:07.970","Text":"In other words, if a function cannot take"},{"Start":"03:07.970 ","End":"03:11.825","Text":"2 different values to the same value, it\u0027s called one-to-one."},{"Start":"03:11.825 ","End":"03:15.515","Text":"In the section on one-to-one functions which hopefully you\u0027ve studied."},{"Start":"03:15.515 ","End":"03:19.915","Text":"In other words, our function f is not one-to-one."},{"Start":"03:19.915 ","End":"03:24.530","Text":"One-to-one means it takes different values for different values and not one-to-one,"},{"Start":"03:24.530 ","End":"03:27.620","Text":"because it takes different values to the same value."},{"Start":"03:27.620 ","End":"03:30.785","Text":"Conclusion I can draw from this is the following."},{"Start":"03:30.785 ","End":"03:32.540","Text":"If a function has an inverse,"},{"Start":"03:32.540 ","End":"03:33.965","Text":"it must be one-to-one."},{"Start":"03:33.965 ","End":"03:38.368","Text":"If your function is not one-to-one or if it takes 2 different values for the same value,"},{"Start":"03:38.368 ","End":"03:39.720","Text":"it doesn\u0027t have an inverse."},{"Start":"03:39.720 ","End":"03:42.320","Text":"The one who writes the exercises usually makes sure that it"},{"Start":"03:42.320 ","End":"03:45.140","Text":"is one-to-one by restricting the domain,"},{"Start":"03:45.140 ","End":"03:46.445","Text":"like in this example."},{"Start":"03:46.445 ","End":"03:49.775","Text":"If I didn\u0027t have the x bigger or equal to 0 here,"},{"Start":"03:49.775 ","End":"03:51.650","Text":"this function is not one-to-one,"},{"Start":"03:51.650 ","End":"03:54.620","Text":"as we can see, it takes both 2 and minus 2-4."},{"Start":"03:54.620 ","End":"03:56.245","Text":"But if we restrict it,"},{"Start":"03:56.245 ","End":"03:58.355","Text":"x being bigger or equal to 0,"},{"Start":"03:58.355 ","End":"04:01.850","Text":"then we wouldn\u0027t have this line at all and that would be okay."},{"Start":"04:01.850 ","End":"04:03.725","Text":"That\u0027s what happens in the exercises."},{"Start":"04:03.725 ","End":"04:07.025","Text":"Domain is restricted and that makes the function one-to-one,"},{"Start":"04:07.025 ","End":"04:09.235","Text":"then it has a chance of having an inverse."},{"Start":"04:09.235 ","End":"04:11.930","Text":"Let\u0027s go to the other remark on the following page."},{"Start":"04:11.930 ","End":"04:15.770","Text":"This time I\u0027m going to take f(x) is equal to square root of x."},{"Start":"04:15.770 ","End":"04:19.305","Text":"Automatically I restrict the domain to x bigger or equal to 0,"},{"Start":"04:19.305 ","End":"04:21.445","Text":"otherwise, this is not defined."},{"Start":"04:21.445 ","End":"04:26.510","Text":"Now, I\u0027m going to try and find the inverse of f. So using our usual methods,"},{"Start":"04:26.510 ","End":"04:31.895","Text":"we just change to the y notation and write y equals the square root of x."},{"Start":"04:31.895 ","End":"04:37.129","Text":"Then we switch between x and y so that x equals"},{"Start":"04:37.129 ","End":"04:39.440","Text":"the square root of y. I\u0027d like to put"},{"Start":"04:39.440 ","End":"04:43.138","Text":"some dotted line here above and below the switch because,"},{"Start":"04:43.138 ","End":"04:44.885","Text":"for example, after the switch,"},{"Start":"04:44.885 ","End":"04:49.730","Text":"this limitation would translate to y bigger or equal to 0,"},{"Start":"04:49.730 ","End":"04:55.100","Text":"trying to isolate y in terms of x. I square both sides,"},{"Start":"04:55.100 ","End":"04:57.140","Text":"I get x^2 equals y,"},{"Start":"04:57.140 ","End":"05:03.640","Text":"which I\u0027ll just write as y equals x^2 and then put it in functional notation."},{"Start":"05:03.640 ","End":"05:08.480","Text":"The inverse function, what it does to x is it turns it into x^2."},{"Start":"05:08.480 ","End":"05:10.325","Text":"So where\u0027s the problem?"},{"Start":"05:10.325 ","End":"05:14.990","Text":"The problem is that this inverse of f doesn\u0027t quite behave like an inverse."},{"Start":"05:14.990 ","End":"05:16.415","Text":"I\u0027ll show you what I mean."},{"Start":"05:16.415 ","End":"05:19.955","Text":"Suppose I start with x equals, say,"},{"Start":"05:19.955 ","End":"05:24.530","Text":"negative 3, and apply the inverse function of f to it."},{"Start":"05:24.530 ","End":"05:28.110","Text":"Well, this is equal to x^2 and I get 9,"},{"Start":"05:28.110 ","End":"05:30.050","Text":"which is negative 3 squared."},{"Start":"05:30.050 ","End":"05:33.890","Text":"And now suppose I want to apply f to 9."},{"Start":"05:33.890 ","End":"05:36.500","Text":"I take 9 and apply f to it."},{"Start":"05:36.500 ","End":"05:38.270","Text":"Then I look at what f is."},{"Start":"05:38.270 ","End":"05:39.500","Text":"F is the square root of x,"},{"Start":"05:39.500 ","End":"05:41.060","Text":"and it goes to 3."},{"Start":"05:41.060 ","End":"05:43.385","Text":"Now, if these were inverses of each other,"},{"Start":"05:43.385 ","End":"05:45.620","Text":"everything should go back to where it came from."},{"Start":"05:45.620 ","End":"05:49.220","Text":"If I apply first f minus 1 and then f to the answer,"},{"Start":"05:49.220 ","End":"05:53.645","Text":"I should get back to where I started from the original x. Here it isn\u0027t."},{"Start":"05:53.645 ","End":"05:55.745","Text":"The other way around works fine."},{"Start":"05:55.745 ","End":"05:59.705","Text":"For example, if I start with 4 and I apply f,"},{"Start":"05:59.705 ","End":"06:02.735","Text":"I\u0027ll get square root of 4, which is 2."},{"Start":"06:02.735 ","End":"06:06.485","Text":"Then if I take 2 and apply the inverse of f,"},{"Start":"06:06.485 ","End":"06:09.400","Text":"which is 2 squared, I get to 4."},{"Start":"06:09.400 ","End":"06:11.855","Text":"So it does go back to where it started from."},{"Start":"06:11.855 ","End":"06:14.000","Text":"But the problem is that in this case,"},{"Start":"06:14.000 ","End":"06:16.000","Text":"if I do first f minus 1 and then f,"},{"Start":"06:16.000 ","End":"06:17.870","Text":"I don\u0027t always get back where I started from."},{"Start":"06:17.870 ","End":"06:20.360","Text":"So this is not really an inverse function."},{"Start":"06:20.360 ","End":"06:24.234","Text":"However, if I added the condition and here\u0027s the condition,"},{"Start":"06:24.234 ","End":"06:26.465","Text":"when I look at this original function f,"},{"Start":"06:26.465 ","End":"06:28.550","Text":"not only is x restricted,"},{"Start":"06:28.550 ","End":"06:29.870","Text":"but if you think about it,"},{"Start":"06:29.870 ","End":"06:32.600","Text":"y is also restricted quite naturally because"},{"Start":"06:32.600 ","End":"06:36.350","Text":"the square root always has to be bigger or equal to 0."},{"Start":"06:36.350 ","End":"06:37.700","Text":"So in this case,"},{"Start":"06:37.700 ","End":"06:42.380","Text":"I automatically get that y has to be bigger or equal to 0."},{"Start":"06:42.380 ","End":"06:44.750","Text":"Then after I switched x and y,"},{"Start":"06:44.750 ","End":"06:49.205","Text":"I have to add the condition that x is bigger or equal to 0."},{"Start":"06:49.205 ","End":"06:51.770","Text":"If I say that the inverse function is this,"},{"Start":"06:51.770 ","End":"06:54.305","Text":"but only for x bigger than 0,"},{"Start":"06:54.305 ","End":"06:55.880","Text":"then everything would be fine."},{"Start":"06:55.880 ","End":"06:58.550","Text":"Because then I couldn\u0027t use this example"},{"Start":"06:58.550 ","End":"07:01.775","Text":"because I could only start with xs which are bigger or equal to 0."},{"Start":"07:01.775 ","End":"07:04.910","Text":"All this relates to the concept of domain,"},{"Start":"07:04.910 ","End":"07:06.730","Text":"an image of a function,"},{"Start":"07:06.730 ","End":"07:09.350","Text":"which is discussed in another clip."},{"Start":"07:09.350 ","End":"07:12.784","Text":"But just notice here that the domain of this function"},{"Start":"07:12.784 ","End":"07:18.020","Text":"was non-negative x bigger or equal to 0 But the image of the function,"},{"Start":"07:18.020 ","End":"07:22.010","Text":"meaning all the possible y\u0027s you can get was bigger or equal to 0."},{"Start":"07:22.010 ","End":"07:23.480","Text":"When you do the switch,"},{"Start":"07:23.480 ","End":"07:26.210","Text":"you have to take the image of the first one and make"},{"Start":"07:26.210 ","End":"07:29.060","Text":"it as the domain of the inverse function."},{"Start":"07:29.060 ","End":"07:32.734","Text":"But as I say, the concept of image is discussed elsewhere."},{"Start":"07:32.734 ","End":"07:35.630","Text":"This is another fine point that we discussed in"},{"Start":"07:35.630 ","End":"07:39.780","Text":"the concept of inverse function. That\u0027s all for now."}],"ID":8406},{"Watched":false,"Name":"The Image of a Function","Duration":"8m 17s","ChapterTopicVideoID":8247,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this clip, I\u0027m going to talk about the concept of the image of a function."},{"Start":"00:04.620 ","End":"00:07.815","Text":"Let\u0027s take the example of a particular function."},{"Start":"00:07.815 ","End":"00:12.555","Text":"In general, a function is y equals f(x)."},{"Start":"00:12.555 ","End":"00:17.235","Text":"And in our case, we\u0027ll take our f(x) to be x^2."},{"Start":"00:17.235 ","End":"00:19.845","Text":"If I ask what is f(3),"},{"Start":"00:19.845 ","End":"00:22.095","Text":"then I\u0027d say 3^2 is 9."},{"Start":"00:22.095 ","End":"00:25.125","Text":"So if x is 3, y is 9."},{"Start":"00:25.125 ","End":"00:32.640","Text":"Often, we describe this with an arrow with take 3 and put a little arrow there and say 3"},{"Start":"00:32.640 ","End":"00:36.480","Text":"goes to 9 with the function f. The function f takes in"},{"Start":"00:36.480 ","End":"00:41.120","Text":"3 as input and gives out 9 as output."},{"Start":"00:41.120 ","End":"00:42.650","Text":"In general, x goes in,"},{"Start":"00:42.650 ","End":"00:46.235","Text":"y goes out, but we could take several examples."},{"Start":"00:46.235 ","End":"00:51.960","Text":"We could say that minus 5 goes to 25."},{"Start":"00:51.960 ","End":"00:56.310","Text":"You could say that 0 goes to 0,"},{"Start":"00:56.310 ","End":"01:00.065","Text":"10 goes to 100, and so on."},{"Start":"01:00.065 ","End":"01:02.600","Text":"Now, I\u0027m going to talk about not only image"},{"Start":"01:02.600 ","End":"01:05.420","Text":"but domain because they have some similar quality."},{"Start":"01:05.420 ","End":"01:08.600","Text":"So if I talk about the domain of a function,"},{"Start":"01:08.600 ","End":"01:09.950","Text":"and in our case,"},{"Start":"01:09.950 ","End":"01:13.070","Text":"this particular function, what I can ask is,"},{"Start":"01:13.070 ","End":"01:16.054","Text":"which values of x are allowed as input."},{"Start":"01:16.054 ","End":"01:17.540","Text":"In general, for a function f,"},{"Start":"01:17.540 ","End":"01:19.594","Text":"that\u0027s how we define domain."},{"Start":"01:19.594 ","End":"01:22.430","Text":"What I mean is, what can I put into this machine"},{"Start":"01:22.430 ","End":"01:25.460","Text":"if you like f or what can it take as input?"},{"Start":"01:25.460 ","End":"01:27.470","Text":"Well, any number could be squared,"},{"Start":"01:27.470 ","End":"01:28.610","Text":"so in this case,"},{"Start":"01:28.610 ","End":"01:31.205","Text":"the answer would be all x."},{"Start":"01:31.205 ","End":"01:34.805","Text":"Now we come to the concept of image."},{"Start":"01:34.805 ","End":"01:37.520","Text":"The image asks the question,"},{"Start":"01:37.520 ","End":"01:41.750","Text":"which values of y can we get as output,"},{"Start":"01:41.750 ","End":"01:45.440","Text":"we\u0027ve moved from the world of x\u0027s to the world of y\u0027s."},{"Start":"01:45.440 ","End":"01:48.005","Text":"Well, there\u0027s 4 examples I can give you already."},{"Start":"01:48.005 ","End":"01:49.640","Text":"9 can be an output,"},{"Start":"01:49.640 ","End":"01:51.455","Text":"25 can be an output,"},{"Start":"01:51.455 ","End":"01:54.235","Text":"0 and 100 can also be."},{"Start":"01:54.235 ","End":"01:59.120","Text":"Then a bit of highlighting where the yellow is in the world"},{"Start":"01:59.120 ","End":"02:04.175","Text":"of the y\u0027s and the turquoise is in the world of x\u0027s."},{"Start":"02:04.175 ","End":"02:09.740","Text":"So domain relates to x\u0027s and asks which x\u0027s are we allowed to"},{"Start":"02:09.740 ","End":"02:16.084","Text":"put into the function and image is all possible y\u0027s that can come out of the function."},{"Start":"02:16.084 ","End":"02:20.290","Text":"It\u0027s not immediately clear which values can come out of f,"},{"Start":"02:20.290 ","End":"02:22.625","Text":"but if we think about it, I can ask,"},{"Start":"02:22.625 ","End":"02:26.615","Text":"is 16 in the image and then I\u0027d have to say yes,"},{"Start":"02:26.615 ","End":"02:33.155","Text":"because 4 goes to 16 by means of f. If I ask if 0 is in the image."},{"Start":"02:33.155 ","End":"02:35.105","Text":"Also, yes, here we\u0027ve seen it."},{"Start":"02:35.105 ","End":"02:36.635","Text":"What about if I ask,"},{"Start":"02:36.635 ","End":"02:39.350","Text":"is 11 in the image also,"},{"Start":"02:39.350 ","End":"02:42.880","Text":"yes, because we can put in the square root of 11."},{"Start":"02:42.880 ","End":"02:44.390","Text":"If I put in as x,"},{"Start":"02:44.390 ","End":"02:45.860","Text":"the square root of 11,"},{"Start":"02:45.860 ","End":"02:48.275","Text":"the y will come out as 11."},{"Start":"02:48.275 ","End":"02:50.480","Text":"It almost looks like everything goes,"},{"Start":"02:50.480 ","End":"02:52.040","Text":"but if I asked you now,"},{"Start":"02:52.040 ","End":"02:54.790","Text":"how about minus 9."},{"Start":"02:54.790 ","End":"02:56.780","Text":"Is minus 9 in the image?"},{"Start":"02:56.780 ","End":"02:58.385","Text":"Is there some value of x,"},{"Start":"02:58.385 ","End":"02:59.780","Text":"which if I put it into f,"},{"Start":"02:59.780 ","End":"03:04.055","Text":"will come out as minus 9 and the answer is no,"},{"Start":"03:04.055 ","End":"03:09.545","Text":"because any x that I put in will come out positive or at least non-negative,"},{"Start":"03:09.545 ","End":"03:12.305","Text":"so I can\u0027t get a negative number out."},{"Start":"03:12.305 ","End":"03:13.460","Text":"When you think about it,"},{"Start":"03:13.460 ","End":"03:18.110","Text":"all positive numbers could be in the image because like the example of 11,"},{"Start":"03:18.110 ","End":"03:19.895","Text":"I could always take its square root."},{"Start":"03:19.895 ","End":"03:22.010","Text":"Square root of 11 would give me 11,"},{"Start":"03:22.010 ","End":"03:25.340","Text":"but no negative value can be obtained because nothing"},{"Start":"03:25.340 ","End":"03:28.970","Text":"squared will give me negative other than 0, it already is."},{"Start":"03:28.970 ","End":"03:30.755","Text":"It looks like 0 is good,"},{"Start":"03:30.755 ","End":"03:33.575","Text":"positive is good, negative is bad."},{"Start":"03:33.575 ","End":"03:41.059","Text":"We can see that the answer to this will be all the y\u0027s which are bigger or equal to 0."},{"Start":"03:41.059 ","End":"03:44.495","Text":"So once again, domain is in the world of x\u0027s,"},{"Start":"03:44.495 ","End":"03:46.760","Text":"which x\u0027s can be put into f,"},{"Start":"03:46.760 ","End":"03:50.750","Text":"image is in the world of y\u0027s and is which y\u0027s can come"},{"Start":"03:50.750 ","End":"03:55.120","Text":"out of f. Now how about the following example."},{"Start":"03:55.120 ","End":"03:58.505","Text":"y, which is some function of x,"},{"Start":"03:58.505 ","End":"04:04.140","Text":"which happens to be 4x plus 1 over x"},{"Start":"04:04.140 ","End":"04:10.895","Text":"minus 10 and I\u0027d like to know what is the domain and what is the image."},{"Start":"04:10.895 ","End":"04:16.460","Text":"Well, the domain is fairly easy to see because you just inspect this."},{"Start":"04:16.460 ","End":"04:20.090","Text":"You\u0027ll see that any value of x except 10 will do,"},{"Start":"04:20.090 ","End":"04:23.740","Text":"so I just say x is not equal to 10."},{"Start":"04:23.740 ","End":"04:25.880","Text":"The image is a bit more difficult."},{"Start":"04:25.880 ","End":"04:27.155","Text":"I mean, how am I going to do this?"},{"Start":"04:27.155 ","End":"04:32.075","Text":"I\u0027m just going to put in a whole bunch of x\u0027s and see what comes out?"},{"Start":"04:32.075 ","End":"04:35.080","Text":"I mean if I say x equals 1,"},{"Start":"04:35.080 ","End":"04:38.550","Text":"I\u0027ll get 5 over minus 9."},{"Start":"04:38.550 ","End":"04:40.110","Text":"If I put x equals 0,"},{"Start":"04:40.110 ","End":"04:43.760","Text":"I\u0027ll get 1 over minus 10 and I\u0027ll collect a bunch of numbers."},{"Start":"04:43.760 ","End":"04:45.530","Text":"I don\u0027t know if it\u0027ll make any sense."},{"Start":"04:45.530 ","End":"04:49.370","Text":"What we need is something more systematic to find the image,"},{"Start":"04:49.370 ","End":"04:54.605","Text":"to see which values of y we can get out of this and I\u0027m going to show you this system."},{"Start":"04:54.605 ","End":"04:56.360","Text":"I\u0027m going to give you a technique,"},{"Start":"04:56.360 ","End":"04:58.475","Text":"but it doesn\u0027t work in all cases."},{"Start":"04:58.475 ","End":"05:00.665","Text":"It works when f has an inverse function."},{"Start":"05:00.665 ","End":"05:02.529","Text":"If we\u0027re lucky enough,"},{"Start":"05:02.529 ","End":"05:05.230","Text":"and most functions will have an inverse that you\u0027ll see."},{"Start":"05:05.230 ","End":"05:08.390","Text":"If f has an inverse,"},{"Start":"05:08.390 ","End":"05:11.855","Text":"share will be f to the minus 1 that\u0027s how it\u0027s written."},{"Start":"05:11.855 ","End":"05:14.690","Text":"Then the image of f is"},{"Start":"05:14.690 ","End":"05:17.720","Text":"exactly the domain of the inverse of"},{"Start":"05:17.720 ","End":"05:21.410","Text":"f. Let\u0027s be optimistic and hope this one has an inverse,"},{"Start":"05:21.410 ","End":"05:24.995","Text":"and we\u0027ll find the inverse first using our usual technique."},{"Start":"05:24.995 ","End":"05:29.240","Text":"So we start off with f(x) equals 4x plus 1 over x minus 10."},{"Start":"05:29.240 ","End":"05:32.975","Text":"That\u0027s already written. Then we replace f(x) with y."},{"Start":"05:32.975 ","End":"05:34.430","Text":"So we have y equals this,"},{"Start":"05:34.430 ","End":"05:35.810","Text":"that was the next stage."},{"Start":"05:35.810 ","End":"05:38.610","Text":"Then we replace x with y and y with x,"},{"Start":"05:38.610 ","End":"05:39.995","Text":"we switch between them."},{"Start":"05:39.995 ","End":"05:48.850","Text":"We get x equals 4y plus 1 over y minus 10."},{"Start":"05:48.850 ","End":"05:51.710","Text":"Next, we try and isolate y"},{"Start":"05:51.710 ","End":"05:55.355","Text":"on the left and everything else with x on the right. Let\u0027s see."},{"Start":"05:55.355 ","End":"06:04.850","Text":"Multiply out and we get x times y minus 10x is equal to 4y plus 1."},{"Start":"06:04.850 ","End":"06:07.115","Text":"Now let\u0027s put everything with y in it."},{"Start":"06:07.115 ","End":"06:09.875","Text":"There\u0027s y here and there\u0027s y here."},{"Start":"06:09.875 ","End":"06:12.395","Text":"Let\u0027s put all those on the left hand side."},{"Start":"06:12.395 ","End":"06:18.050","Text":"We get xy minus 4y equals,"},{"Start":"06:18.050 ","End":"06:22.295","Text":"and then the x is on the right, 10x plus 1."},{"Start":"06:22.295 ","End":"06:24.304","Text":"If we take y outside the brackets."},{"Start":"06:24.304 ","End":"06:28.340","Text":"We have x minus 4 in brackets and the x minus 4,"},{"Start":"06:28.340 ","End":"06:31.490","Text":"I can bring straightaway onto the denominator,"},{"Start":"06:31.490 ","End":"06:34.035","Text":"and this is left there."},{"Start":"06:34.035 ","End":"06:42.595","Text":"Finally, we write that the inverse function is 10x plus 1 over x minus 4."},{"Start":"06:42.595 ","End":"06:51.530","Text":"What we get is that the domain of f inverse is x not equal to 4."},{"Start":"06:51.530 ","End":"06:55.024","Text":"This means, according to the theorem,"},{"Start":"06:55.024 ","End":"06:59.480","Text":"that the image of f itself is the same"},{"Start":"06:59.480 ","End":"07:04.370","Text":"except that the image is in the world of y not the world of x."},{"Start":"07:04.370 ","End":"07:11.075","Text":"The image of f is y not equal to 4 and that\u0027s the answer."},{"Start":"07:11.075 ","End":"07:17.315","Text":"What this means is that I can never get y equals 4 in terms of f(x),"},{"Start":"07:17.315 ","End":"07:19.325","Text":"no matter what x I put in here,"},{"Start":"07:19.325 ","End":"07:21.080","Text":"I will never get y equals 4."},{"Start":"07:21.080 ","End":"07:23.600","Text":"You can try putting in values all you like."},{"Start":"07:23.600 ","End":"07:25.430","Text":"You can\u0027t get 4."},{"Start":"07:25.430 ","End":"07:29.180","Text":"It might even try and get clever and write an equation that this is"},{"Start":"07:29.180 ","End":"07:33.065","Text":"equal to 4 and you\u0027ll see that this equation has no solution."},{"Start":"07:33.065 ","End":"07:36.515","Text":"To review the technique of finding the image,"},{"Start":"07:36.515 ","End":"07:41.180","Text":"which this question mark I can now replace by a real answer,"},{"Start":"07:41.180 ","End":"07:44.610","Text":"y is not equal to 4."},{"Start":"07:44.610 ","End":"07:46.665","Text":"So again, going over the technique,"},{"Start":"07:46.665 ","End":"07:51.620","Text":"we have our function and we use the technique of finding an inverse function,"},{"Start":"07:51.620 ","End":"07:53.735","Text":"replacing y with x,"},{"Start":"07:53.735 ","End":"07:56.824","Text":"solving it, getting what y equals,"},{"Start":"07:56.824 ","End":"08:01.354","Text":"then finding the domain of this inverse function,"},{"Start":"08:01.354 ","End":"08:02.945","Text":"finding what it is in this case,"},{"Start":"08:02.945 ","End":"08:05.390","Text":"x not equal to 4 and finally,"},{"Start":"08:05.390 ","End":"08:07.160","Text":"substituting instead of x,"},{"Start":"08:07.160 ","End":"08:10.610","Text":"we put y because the image lives in the world of y,"},{"Start":"08:10.610 ","End":"08:12.950","Text":"whereas the domain lives in the world of x."},{"Start":"08:12.950 ","End":"08:15.080","Text":"And that\u0027s basically all there is to it."},{"Start":"08:15.080 ","End":"08:17.700","Text":"I\u0027m done with this clip."}],"ID":8407},{"Watched":false,"Name":"Exercise 1","Duration":"12m 34s","ChapterTopicVideoID":4507,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Here we have 3 exercises, really a, b,"},{"Start":"00:03.330 ","End":"00:05.040","Text":"and c. In each one,"},{"Start":"00:05.040 ","End":"00:06.510","Text":"we\u0027re given f(x),"},{"Start":"00:06.510 ","End":"00:09.090","Text":"we have to find the inverse function f^minus 1"},{"Start":"00:09.090 ","End":"00:14.100","Text":"and also the image of the function f and finally,"},{"Start":"00:14.100 ","End":"00:16.935","Text":"to show that this equality holds."},{"Start":"00:16.935 ","End":"00:20.320","Text":"Let\u0027s begin with a,"},{"Start":"00:21.230 ","End":"00:25.725","Text":"and the usual technique is to"},{"Start":"00:25.725 ","End":"00:33.330","Text":"switch y with x and x with y and that helps us to find the inverse."},{"Start":"00:33.330 ","End":"00:39.855","Text":"We\u0027ll rewrite a as instead of y=4 natural log x."},{"Start":"00:39.855 ","End":"00:46.755","Text":"We\u0027ll write x=4 natural log of y."},{"Start":"00:46.755 ","End":"00:49.430","Text":"We have to extract y in terms of x,"},{"Start":"00:49.430 ","End":"00:53.105","Text":"and this trick will give us the inverse. Let\u0027s see."},{"Start":"00:53.105 ","End":"00:56.095","Text":"We\u0027ll divide both sides by 4,"},{"Start":"00:56.095 ","End":"01:04.730","Text":"x over 4= natural log of y."},{"Start":"01:04.730 ","End":"01:09.300","Text":"We raise each side as an exponent,"},{"Start":"01:09.300 ","End":"01:11.130","Text":"e to the power of each side,"},{"Start":"01:11.130 ","End":"01:14.100","Text":"so we get e^x over 4."},{"Start":"01:14.100 ","End":"01:15.765","Text":"Let\u0027s also switch sides."},{"Start":"01:15.765 ","End":"01:21.160","Text":"E^x over 4=e to the power of natural log,"},{"Start":"01:21.160 ","End":"01:25.885","Text":"the e and the natural log cancel each other out and we get just y."},{"Start":"01:25.885 ","End":"01:33.385","Text":"Or in general, if a equals natural log of b,"},{"Start":"01:33.385 ","End":"01:37.690","Text":"that\u0027s the same thing as b=e^a."},{"Start":"01:37.690 ","End":"01:40.880","Text":"That\u0027s the definition really of the natural log."},{"Start":"01:41.510 ","End":"01:46.395","Text":"I\u0027ll just dispense with that. Let\u0027s continue."},{"Start":"01:46.395 ","End":"01:55.980","Text":"This will be inverse function and I will write it"},{"Start":"01:55.980 ","End":"02:05.085","Text":"as f^minus 1(x) equals e^x over 4."},{"Start":"02:05.085 ","End":"02:08.040","Text":"That\u0027s one part of the question."},{"Start":"02:08.040 ","End":"02:10.730","Text":"Highlight that."},{"Start":"02:10.730 ","End":"02:13.270","Text":"To find the image of f,"},{"Start":"02:13.270 ","End":"02:18.070","Text":"we need to look at the domain of f^minus 1. Let\u0027s see."},{"Start":"02:18.070 ","End":"02:26.465","Text":"What is the domain of f^minus 1."},{"Start":"02:26.465 ","End":"02:30.880","Text":"Just looking at this, this thing is defined for all x."},{"Start":"02:30.880 ","End":"02:35.155","Text":"We just write all x."},{"Start":"02:35.155 ","End":"02:42.690","Text":"That\u0027s the same as the image of f,"},{"Start":"02:42.690 ","End":"02:47.880","Text":"except that in this case we revert back to y."},{"Start":"02:47.880 ","End":"02:50.385","Text":"This is all y."},{"Start":"02:50.385 ","End":"02:53.685","Text":"That\u0027s another part of the question."},{"Start":"02:53.685 ","End":"02:56.535","Text":"I\u0027ll highlight that also."},{"Start":"02:56.535 ","End":"03:01.990","Text":"Now, the final part is to show this equality."},{"Start":"03:02.150 ","End":"03:04.215","Text":"I\u0027ll do it over here."},{"Start":"03:04.215 ","End":"03:11.005","Text":"Let\u0027s see f(f)^minus 1(x)."},{"Start":"03:11.005 ","End":"03:15.080","Text":"Let\u0027s do a series of steps and hope we get to x in the end."},{"Start":"03:15.080 ","End":"03:17.126","Text":"This is f of, now,"},{"Start":"03:17.126 ","End":"03:22.400","Text":"f^minus 1(x) is e^x over 4."},{"Start":"03:22.400 ","End":"03:25.670","Text":"This is equal to, using the definition of f,"},{"Start":"03:25.670 ","End":"03:35.510","Text":"which is 4 natural log (x)=4 natural log (e^x over 4),"},{"Start":"03:35.510 ","End":"03:40.358","Text":"which equals, and the natural log and the exponent cancel each other out,"},{"Start":"03:40.358 ","End":"03:45.340","Text":"so it\u0027s just x over 4, which equals x."},{"Start":"03:45.340 ","End":"03:46.980","Text":"That answers the last part."},{"Start":"03:46.980 ","End":"03:54.075","Text":"We\u0027re done with a. Now, let\u0027s go on to part b."},{"Start":"03:54.075 ","End":"03:57.510","Text":"In b, same as in a."},{"Start":"03:57.510 ","End":"03:59.475","Text":"Using the same techniques,"},{"Start":"03:59.475 ","End":"04:04.755","Text":"we\u0027ll replace y with x and x with y."},{"Start":"04:04.755 ","End":"04:07.260","Text":"This will help us find the inverse function."},{"Start":"04:07.260 ","End":"04:10.905","Text":"Instead of y equals this thing with x,"},{"Start":"04:10.905 ","End":"04:18.991","Text":"I\u0027ll write x=2 plus 3 natural log (y minus 1)."},{"Start":"04:18.991 ","End":"04:23.280","Text":"The idea is to extract y in terms of x."},{"Start":"04:23.800 ","End":"04:31.070","Text":"Let\u0027s bring the 2 to the other side and switch sides."},{"Start":"04:31.070 ","End":"04:36.871","Text":"I\u0027ve got 3 natural log (y minus 1)=x minus 2 over"},{"Start":"04:36.871 ","End":"04:41.880","Text":"3 natural log (y minus"},{"Start":"04:41.880 ","End":"04:48.940","Text":"1)=x minus 2 over 3."},{"Start":"04:52.040 ","End":"04:56.050","Text":"Then if this is the natural log of this and this is"},{"Start":"04:56.050 ","End":"05:00.550","Text":"the exponent of this so y minus 1=e^x minus 2 over 3."},{"Start":"05:05.060 ","End":"05:15.585","Text":"Finally, we get y=1 plus e^(x)2 over 3."},{"Start":"05:15.585 ","End":"05:19.680","Text":"This will be our f minus (1)x,"},{"Start":"05:19.680 ","End":"05:24.490","Text":"which I shall highlight. There we go."},{"Start":"05:24.490 ","End":"05:27.715","Text":"Now we have to identify the image."},{"Start":"05:27.715 ","End":"05:29.590","Text":"To find the image of f,"},{"Start":"05:29.590 ","End":"05:33.865","Text":"we ask what is the domain of the inverse function?"},{"Start":"05:33.865 ","End":"05:37.555","Text":"The domain of f^minus 1. Well, let\u0027s look at it."},{"Start":"05:37.555 ","End":"05:44.455","Text":"What could possibly be a bad value of x to try and substitute."},{"Start":"05:44.455 ","End":"05:46.540","Text":"For any x, we could subtract 2."},{"Start":"05:46.540 ","End":"05:48.925","Text":"Any number can be divided by 3."},{"Start":"05:48.925 ","End":"05:52.585","Text":"Any number we can raise e to the power of. We can always add 1."},{"Start":"05:52.585 ","End":"05:59.500","Text":"Any x can be substituted here so the answer to this one is all x."},{"Start":"05:59.500 ","End":"06:03.620","Text":"When we take the image of f,"},{"Start":"06:06.020 ","End":"06:11.930","Text":"it will be not all x and we have to replace it back with y."},{"Start":"06:12.180 ","End":"06:15.505","Text":"That answers the next bit."},{"Start":"06:15.505 ","End":"06:25.165","Text":"Finally, we want to show that this equality holds this identity."},{"Start":"06:25.165 ","End":"06:28.900","Text":"See if I have room here. Go for it."},{"Start":"06:28.900 ","End":"06:37.655","Text":"f(f)^minus 1, which is the inverse of f (x), is equal to."},{"Start":"06:37.655 ","End":"06:40.270","Text":"First of all, I work from the inside out."},{"Start":"06:40.270 ","End":"06:44.590","Text":"First of all is the inverse which I take from here,"},{"Start":"06:44.590 ","End":"06:51.073","Text":"so this is f(1"},{"Start":"06:51.073 ","End":"06:55.365","Text":"plus e^x minus 2 over 3) which equals,"},{"Start":"06:55.365 ","End":"06:57.105","Text":"now applying f,"},{"Start":"06:57.105 ","End":"07:00.930","Text":"and f is what\u0027s written here."},{"Start":"07:00.930 ","End":"07:09.510","Text":"It\u0027s 2 plus 3 natural log, and instead of x,"},{"Start":"07:09.510 ","End":"07:15.040","Text":"I need to put this whole expression so it\u0027s"},{"Start":"07:23.540 ","End":"07:26.360","Text":"1 plus e^x minus 2 over 3. I\u0027ll put it in brackets for"},{"Start":"07:26.360 ","End":"07:29.585","Text":"emphasis that this takes the place of x here."},{"Start":"07:29.585 ","End":"07:31.885","Text":"Minus 1."},{"Start":"07:31.885 ","End":"07:35.695","Text":"It looks a mess, it\u0027ll soon straighten itself out."},{"Start":"07:35.695 ","End":"07:41.530","Text":"This is equal to 2 plus 3 natural log."},{"Start":"07:41.530 ","End":"07:45.190","Text":"Now, the one with the minus 1 cancel."},{"Start":"07:45.190 ","End":"07:51.110","Text":"All I\u0027m left with is e^x minus 2 over 3."},{"Start":"07:52.670 ","End":"07:58.375","Text":"This is equal to natural log of an exponent is just what\u0027s here."},{"Start":"07:58.375 ","End":"08:01.490","Text":"It\u0027s 2 plus 3."},{"Start":"08:06.350 ","End":"08:12.769","Text":"This equals 3 times 3, this cancels out."},{"Start":"08:12.769 ","End":"08:15.365","Text":"Running out of space here."},{"Start":"08:15.365 ","End":"08:18.350","Text":"Just 2 plus 3s cancel,"},{"Start":"08:18.350 ","End":"08:21.810","Text":"so it\u0027s x minus 2=x."},{"Start":"08:24.440 ","End":"08:27.140","Text":"We\u0027ve shown this because you see,"},{"Start":"08:27.140 ","End":"08:31.370","Text":"we started with this and we ended with this and that\u0027s what we had to do."},{"Start":"08:31.370 ","End":"08:33.845","Text":"Onto the next question."},{"Start":"08:33.845 ","End":"08:39.365","Text":"Now we\u0027ve reached part C, same technique."},{"Start":"08:39.365 ","End":"08:42.631","Text":"Just go with a quicker pace now."},{"Start":"08:42.631 ","End":"08:45.080","Text":"We take this y equals this,"},{"Start":"08:45.080 ","End":"08:50.370","Text":"and replace x with y. x=1 plus 2e^2y."},{"Start":"08:51.430 ","End":"08:57.295","Text":"Now try and isolate y. Subtract 1, x minus 1."},{"Start":"08:57.295 ","End":"09:03.065","Text":"We\u0027ll divide by 2 all in one go equals e^2y."},{"Start":"09:03.065 ","End":"09:05.885","Text":"Take the natural log of both sides,"},{"Start":"09:05.885 ","End":"09:12.450","Text":"that gets rid of the e. Sorry,"},{"Start":"09:13.030 ","End":"09:21.890","Text":"natural log of x minus 1 over 2 equals to y."},{"Start":"09:21.890 ","End":"09:32.480","Text":"Then I\u0027ll switch sides also, y=1/2 natural log (x minus 1 over 2)."},{"Start":"09:32.480 ","End":"09:42.480","Text":"This will be my f minus 1(x)."},{"Start":"09:42.480 ","End":"09:45.325","Text":"That\u0027s the first bit we found f minus 1(x)."},{"Start":"09:45.325 ","End":"09:47.920","Text":"Now, the image."},{"Start":"09:47.920 ","End":"09:49.030","Text":"For the image of f,"},{"Start":"09:49.030 ","End":"09:52.330","Text":"we need the domain of f^minus 1,"},{"Start":"09:52.330 ","End":"09:58.760","Text":"the inverse, the domain of f minus 1."},{"Start":"09:59.550 ","End":"10:02.050","Text":"Let\u0027s see what could that be?"},{"Start":"10:02.050 ","End":"10:06.820","Text":"In other words, what are the bad values of x or the good values of x?"},{"Start":"10:06.820 ","End":"10:13.930","Text":"The only thing to worry about is that the natural log has to have a positive arguments."},{"Start":"10:13.930 ","End":"10:16.630","Text":"I\u0027ll just do this at the sides."},{"Start":"10:16.630 ","End":"10:23.625","Text":"So what we have is x minus 1 over 2 has got to be strictly positive."},{"Start":"10:23.625 ","End":"10:27.785","Text":"Which means that x minus 1 has to also be positive."},{"Start":"10:27.785 ","End":"10:29.990","Text":"Because if something over 2 is positive,"},{"Start":"10:29.990 ","End":"10:33.785","Text":"it has to be positive so x has to be bigger than 1."},{"Start":"10:33.785 ","End":"10:37.345","Text":"So the domain is x bigger than 1?"},{"Start":"10:37.345 ","End":"10:43.790","Text":"This means that the image of f(x)is the same thing,"},{"Start":"10:43.790 ","End":"10:46.880","Text":"but we have to revert back to y."},{"Start":"10:46.880 ","End":"10:50.940","Text":"Is y bigger than 1?"},{"Start":"10:52.550 ","End":"10:56.495","Text":"That answers that part."},{"Start":"10:56.495 ","End":"10:59.540","Text":"Now finally, we have to show this."},{"Start":"10:59.540 ","End":"11:04.324","Text":"I\u0027ll start with the left-hand side and see if I can get to the right hand side."},{"Start":"11:04.324 ","End":"11:11.515","Text":"f(f minus 1) of x."},{"Start":"11:11.515 ","End":"11:13.550","Text":"Let\u0027s see what this equals."},{"Start":"11:13.550 ","End":"11:19.680","Text":"This equals f of f minus 1(x) is written here."},{"Start":"11:23.350 ","End":"11:27.980","Text":"1/2 natural log (x minus 1 over 2). This equals now f. We have"},{"Start":"11:27.980 ","End":"11:31.910","Text":"the formula for f. It\u0027s this, 1 plus 2e^2x."},{"Start":"11:31.910 ","End":"11:37.760","Text":"This equals 1 plus twice e^2."},{"Start":"11:37.760 ","End":"11:41.585","Text":"Instead of x, I\u0027ll put this whole thing here."},{"Start":"11:41.585 ","End":"11:50.600","Text":"It\u0027s 2 times a 1/2 times natural log of x minus 1 over 2."},{"Start":"11:50.600 ","End":"11:53.360","Text":"Let\u0027s see how we can simplify this."},{"Start":"11:53.360 ","End":"12:00.125","Text":"This equals 1 plus twice e^2 with the 1/2 cancels."},{"Start":"12:00.125 ","End":"12:05.150","Text":"That\u0027s just the natural log of x minus 1 over 2."},{"Start":"12:05.150 ","End":"12:06.680","Text":"We\u0027ve seen this before."},{"Start":"12:06.680 ","End":"12:09.470","Text":"When you take e to the power of natural log,"},{"Start":"12:09.470 ","End":"12:11.435","Text":"they cancel each other out."},{"Start":"12:11.435 ","End":"12:17.582","Text":"We just left with x minus 1 over 2."},{"Start":"12:17.582 ","End":"12:20.209","Text":"Now this is equal to the 2 with the 2 cancels,"},{"Start":"12:20.209 ","End":"12:23.345","Text":"it\u0027s x plus 1."},{"Start":"12:23.345 ","End":"12:29.615","Text":"This is equal to x. So you see we started off with this and ended up with this,"},{"Start":"12:29.615 ","End":"12:33.840","Text":"which proves this. We\u0027re done."}],"ID":4516},{"Watched":false,"Name":"Exercise 2","Duration":"2m 29s","ChapterTopicVideoID":4508,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this exercise, we have to find the inverse function of this function,"},{"Start":"00:04.185 ","End":"00:07.665","Text":"f(x), which is log to the base 2 of this expression."},{"Start":"00:07.665 ","End":"00:10.380","Text":"The domain is x bigger than 1,"},{"Start":"00:10.380 ","End":"00:12.525","Text":"and when x is bigger than 1,"},{"Start":"00:12.525 ","End":"00:14.040","Text":"there\u0027s certainly not 0,"},{"Start":"00:14.040 ","End":"00:15.945","Text":"so 1/x is okay."},{"Start":"00:15.945 ","End":"00:18.540","Text":"Also, this thing will be positive."},{"Start":"00:18.540 ","End":"00:20.850","Text":"I could have even written x bigger than 0 and it"},{"Start":"00:20.850 ","End":"00:23.475","Text":"would still be positive and everything would be okay."},{"Start":"00:23.475 ","End":"00:25.830","Text":"Anyway, let\u0027s let y=f(x),"},{"Start":"00:25.830 ","End":"00:27.705","Text":"so y is equal to this."},{"Start":"00:27.705 ","End":"00:29.700","Text":"Now, to find the inverse function,"},{"Start":"00:29.700 ","End":"00:34.210","Text":"one way of doing it is to get x in terms of y. Step-by-step."},{"Start":"00:34.210 ","End":"00:36.540","Text":"Let\u0027s first of all get rid of the log to the base 2,"},{"Start":"00:36.540 ","End":"00:39.425","Text":"and this is what we get just using the rules of logarithms."},{"Start":"00:39.425 ","End":"00:41.330","Text":"The log to the base 2 of this is this,"},{"Start":"00:41.330 ","End":"00:43.475","Text":"then 2 to the this is this, anyway."},{"Start":"00:43.475 ","End":"00:45.830","Text":"Now we want to get rid of the denominator here,"},{"Start":"00:45.830 ","End":"00:48.530","Text":"multiply everything by x, and I get this."},{"Start":"00:48.530 ","End":"00:51.635","Text":"You can see it\u0027s a quadratic equation in x,"},{"Start":"00:51.635 ","End":"00:53.950","Text":"especially when I rewrite it like this."},{"Start":"00:53.950 ","End":"00:58.260","Text":"I\u0027m going to solve this quadratic equation using the quadratic formula,"},{"Start":"00:58.260 ","End":"00:59.990","Text":"and this is what the formula gives."},{"Start":"00:59.990 ","End":"01:02.765","Text":"The trouble is that there\u0027s x_1 and x_2,"},{"Start":"01:02.765 ","End":"01:04.870","Text":"where we can take the plus or we can take the minus,"},{"Start":"01:04.870 ","End":"01:06.515","Text":"and that wouldn\u0027t be a function."},{"Start":"01:06.515 ","End":"01:09.650","Text":"However, because x is bigger than 1,"},{"Start":"01:09.650 ","End":"01:12.200","Text":"it turns out that only the plus is a solution."},{"Start":"01:12.200 ","End":"01:14.240","Text":"I\u0027m not going to go into all the algebra,"},{"Start":"01:14.240 ","End":"01:15.320","Text":"but if I take the minus,"},{"Start":"01:15.320 ","End":"01:16.910","Text":"it\u0027s not going to be bigger than 1,"},{"Start":"01:16.910 ","End":"01:19.760","Text":"and so we now have x in terms of y."},{"Start":"01:19.760 ","End":"01:22.430","Text":"Now, this actually gives us the inverse function,"},{"Start":"01:22.430 ","End":"01:23.930","Text":"but not quite in the way we want."},{"Start":"01:23.930 ","End":"01:27.695","Text":"It gives us x is the inverse function of y,"},{"Start":"01:27.695 ","End":"01:31.595","Text":"but we want to write the inverse function as y in terms of x,"},{"Start":"01:31.595 ","End":"01:34.020","Text":"I mean the latter as a dummy variable."},{"Start":"01:34.120 ","End":"01:36.365","Text":"Perhaps I\u0027ll write that here."},{"Start":"01:36.365 ","End":"01:38.690","Text":"We switched x and y to get"},{"Start":"01:38.690 ","End":"01:42.230","Text":"the function because the function doesn\u0027t depend on the variable."},{"Start":"01:42.230 ","End":"01:43.460","Text":"This is x as a function of y,"},{"Start":"01:43.460 ","End":"01:44.990","Text":"so this is the function of x."},{"Start":"01:44.990 ","End":"01:46.790","Text":"You replace y by x."},{"Start":"01:46.790 ","End":"01:51.785","Text":"The last thing I want us to relate to the domain of the function,"},{"Start":"01:51.785 ","End":"01:54.545","Text":"when is this f to the minus 1 defined?"},{"Start":"01:54.545 ","End":"01:58.715","Text":"Well, what\u0027s under the square root sign has to be non-negative."},{"Start":"01:58.715 ","End":"02:03.005","Text":"This, what\u0027s under the square root has to be bigger or equal to 0 for the domain."},{"Start":"02:03.005 ","End":"02:04.400","Text":"Now, this is not hard to solve,"},{"Start":"02:04.400 ","End":"02:06.380","Text":"just bring the 4 to the other side."},{"Start":"02:06.380 ","End":"02:08.855","Text":"Notice that 4 is 2^2,"},{"Start":"02:08.855 ","End":"02:11.570","Text":"and now I have 2 things equal with equal bases."},{"Start":"02:11.570 ","End":"02:13.575","Text":"There\u0027s a 2 here and a 2 here,"},{"Start":"02:13.575 ","End":"02:16.490","Text":"and so since the base is bigger than 1,"},{"Start":"02:16.490 ","End":"02:18.546","Text":"it\u0027s an increasing function."},{"Start":"02:18.546 ","End":"02:20.270","Text":"So 2x is bigger than 2,"},{"Start":"02:20.270 ","End":"02:22.715","Text":"and finally, x bigger or equal to 1."},{"Start":"02:22.715 ","End":"02:29.910","Text":"To summarize, we found the inverse function here and its domain is here, and we\u0027re done."}],"ID":4517},{"Watched":false,"Name":"Exercise 3 part a","Duration":"3m 9s","ChapterTopicVideoID":4688,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"This exercise has several parts; a, b,"},{"Start":"00:02.400 ","End":"00:06.645","Text":"and c. In each case we\u0027re given a function y= f(x),"},{"Start":"00:06.645 ","End":"00:13.545","Text":"we have to find the inverse function f minus 1(x) to identify the image."},{"Start":"00:13.545 ","End":"00:20.460","Text":"Finally, as a check to show that f(f) minus 1(x) is equal to x."},{"Start":"00:20.460 ","End":"00:23.070","Text":"Let\u0027s begin with part a."},{"Start":"00:23.070 ","End":"00:31.755","Text":"The way to find f minus 1(x) is to take the function and substitute y for x and x for y."},{"Start":"00:31.755 ","End":"00:34.550","Text":"What we get here is the same thing,"},{"Start":"00:34.550 ","End":"00:44.530","Text":"but with the variables that replaced x equals 2y plus 1 over 4y minus 2."},{"Start":"00:44.530 ","End":"00:50.255","Text":"Now we have to do some algebra to extract what y is in terms of x."},{"Start":"00:50.255 ","End":"00:57.965","Text":"Multiply both sides by 4y minus 2 to get rid of the denominator and we get"},{"Start":"00:57.965 ","End":"01:07.855","Text":"4yx minus 2x is equal to 2y plus 1."},{"Start":"01:07.855 ","End":"01:14.245","Text":"Next thing to do is to bring all the y\u0027s to the left and the rest to the right hand side."},{"Start":"01:14.245 ","End":"01:23.230","Text":"We get 4yx minus 2y equals 2x plus 1."},{"Start":"01:23.230 ","End":"01:26.090","Text":"Taking y outside the brackets here,"},{"Start":"01:26.090 ","End":"01:35.860","Text":"we get that 4x minus 2 times y equals 2x plus 1."},{"Start":"01:35.860 ","End":"01:39.075","Text":"Finally, isolating y,"},{"Start":"01:39.075 ","End":"01:46.690","Text":"we get that y equals 2x plus 1 over 4x minus 2."},{"Start":"01:46.690 ","End":"01:53.940","Text":"This is what we call the inverse function of x, f minus 1(x)."},{"Start":"01:54.010 ","End":"01:56.180","Text":"We found this part."},{"Start":"01:56.180 ","End":"02:00.860","Text":"The next thing we have to do is to find the image of"},{"Start":"02:00.860 ","End":"02:06.455","Text":"f. The image of f is simply the domain of f minus 1(x),"},{"Start":"02:06.455 ","End":"02:08.675","Text":"so we have to ask ourselves,"},{"Start":"02:08.675 ","End":"02:14.780","Text":"what is the domain of this in order to get the image of f?"},{"Start":"02:14.780 ","End":"02:18.360","Text":"f minus 1(x) is this."},{"Start":"02:18.360 ","End":"02:20.450","Text":"If we look at it, the domain,"},{"Start":"02:20.450 ","End":"02:23.330","Text":"which means the values of x we\u0027re allowed to substitute,"},{"Start":"02:23.330 ","End":"02:29.105","Text":"it could be anything except that we have to guarantee that the denominator is not 0."},{"Start":"02:29.105 ","End":"02:36.920","Text":"The denominator is not 0 means that 4x minus 2 not equal to 0."},{"Start":"02:36.920 ","End":"02:39.170","Text":"The tiny bit of algebra,"},{"Start":"02:39.170 ","End":"02:44.255","Text":"we can see that this means that x is not equal to 2 over 4,"},{"Start":"02:44.255 ","End":"02:47.015","Text":"x is not equal to 1/2."},{"Start":"02:47.015 ","End":"02:51.290","Text":"However, domains are usually written in terms of the letter x,"},{"Start":"02:51.290 ","End":"02:54.200","Text":"but the image is usually written in terms of y."},{"Start":"02:54.200 ","End":"03:01.210","Text":"Really we should say that the image is y not equal to 1/2."},{"Start":"03:01.210 ","End":"03:03.660","Text":"This is the domain of f minus 1,"},{"Start":"03:03.660 ","End":"03:09.610","Text":"but this is the image of f. We\u0027re done for part a."}],"ID":4696},{"Watched":false,"Name":"Exercise 3 part b","Duration":"2m 27s","ChapterTopicVideoID":4679,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"Next, Part B, as before,"},{"Start":"00:03.360 ","End":"00:08.370","Text":"we reverse the roles of y and x in order to find the inverse function,"},{"Start":"00:08.370 ","End":"00:13.675","Text":"and we write x equals 1/y^3."},{"Start":"00:13.675 ","End":"00:16.155","Text":"From here with just switching variables,"},{"Start":"00:16.155 ","End":"00:17.880","Text":"we\u0027re going to extract y,"},{"Start":"00:17.880 ","End":"00:23.330","Text":"so y^3 is 1/x by simply switching these 2"},{"Start":"00:23.330 ","End":"00:30.045","Text":"around and y is equal to the cube root of 1/x."},{"Start":"00:30.045 ","End":"00:36.525","Text":"This is the answer for the inverse function f minus 1(x)."},{"Start":"00:36.525 ","End":"00:40.560","Text":"Now, to find the image of the function f,"},{"Start":"00:40.560 ","End":"00:47.510","Text":"we need to ask what is the domain of f minus 1(x) and at the end,"},{"Start":"00:47.510 ","End":"00:50.185","Text":"we just write y instead of x."},{"Start":"00:50.185 ","End":"00:53.565","Text":"We need to find the domain of this function here,"},{"Start":"00:53.565 ","End":"00:57.290","Text":"x obviously can\u0027t be 0 because we can\u0027t divide by 0."},{"Start":"00:57.290 ","End":"00:59.690","Text":"But other than that, x could be anything."},{"Start":"00:59.690 ","End":"01:02.210","Text":"Because here we have a cube root, not a square root,"},{"Start":"01:02.210 ","End":"01:05.120","Text":"and the cube root is okay for positive and negative."},{"Start":"01:05.120 ","End":"01:08.690","Text":"The only restriction is that x should not equal to"},{"Start":"01:08.690 ","End":"01:12.410","Text":"0 that\u0027s as far as the domain of f minus 1."},{"Start":"01:12.410 ","End":"01:13.700","Text":"But when we write the image,"},{"Start":"01:13.700 ","End":"01:15.250","Text":"we use the letter y,"},{"Start":"01:15.250 ","End":"01:19.790","Text":"so the image is y not equal to 0;"},{"Start":"01:19.790 ","End":"01:24.245","Text":"this is the image of the function f. Finally,"},{"Start":"01:24.245 ","End":"01:30.185","Text":"we have to check that this identity holds that f(f)minus 1(x) is x,"},{"Start":"01:30.185 ","End":"01:33.270","Text":"so let\u0027s do that,"},{"Start":"01:33.340 ","End":"01:39.380","Text":"f(f) minus 1(x) is equal to f (f)"},{"Start":"01:39.380 ","End":"01:46.200","Text":"minus 1(x) is the cube root of 1/x."},{"Start":"01:46.200 ","End":"01:50.105","Text":"Remember the definition of f, in this case,"},{"Start":"01:50.105 ","End":"01:54.955","Text":"this is our f(x) here is 1/x^3,"},{"Start":"01:54.955 ","End":"02:02.070","Text":"so this is equal to 1 over this thing cubed."},{"Start":"02:02.070 ","End":"02:06.100","Text":"So it\u0027s the cube root of 1/x^3."},{"Start":"02:06.470 ","End":"02:08.970","Text":"Now, if you take the cube root,"},{"Start":"02:08.970 ","End":"02:12.120","Text":"and you cube it, you\u0027re back to the original thing."},{"Start":"02:12.120 ","End":"02:17.730","Text":"The original is 1/x and 1/1/x is simply"},{"Start":"02:17.730 ","End":"02:24.165","Text":"x and this is what we had to show that f(f) minus 1(x) is x."},{"Start":"02:24.165 ","End":"02:27.250","Text":"We\u0027re done with Part B."}],"ID":4688},{"Watched":false,"Name":"Exercise 3 part c","Duration":"2m 11s","ChapterTopicVideoID":4682,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"The last one is part c,"},{"Start":"00:02.475 ","End":"00:07.020","Text":"which I prefer to write as y equals x squared."},{"Start":"00:07.020 ","End":"00:08.985","Text":"We use our usual technique,"},{"Start":"00:08.985 ","End":"00:10.770","Text":"finding the inverse function."},{"Start":"00:10.770 ","End":"00:12.255","Text":"Oh, and let me add that."},{"Start":"00:12.255 ","End":"00:14.730","Text":"This does have an inverse function because it\u0027s"},{"Start":"00:14.730 ","End":"00:17.535","Text":"one-to-one and this was in a previous exercise."},{"Start":"00:17.535 ","End":"00:21.630","Text":"Usual technique of switching y and x,"},{"Start":"00:21.630 ","End":"00:26.985","Text":"we get that x=y^2."},{"Start":"00:26.985 ","End":"00:29.760","Text":"But also the limitation,"},{"Start":"00:29.760 ","End":"00:34.215","Text":"we also switched letters y bigger or equal to 0."},{"Start":"00:34.215 ","End":"00:36.660","Text":"If we switch sides here,"},{"Start":"00:36.660 ","End":"00:41.690","Text":"we get that y squared equals x and y you"},{"Start":"00:41.690 ","End":"00:47.330","Text":"might want to say is equal to plus or minus the square root of x."},{"Start":"00:47.330 ","End":"00:50.930","Text":"But because y is bigger or equal to 0,"},{"Start":"00:50.930 ","End":"00:55.925","Text":"we don\u0027t need the plus or minus and just simply going to remove it."},{"Start":"00:55.925 ","End":"00:59.180","Text":"Y equals the square root of x,"},{"Start":"00:59.180 ","End":"01:01.370","Text":"and this is our inverse function,"},{"Start":"01:01.370 ","End":"01:04.770","Text":"f to the minus 1 of x."},{"Start":"01:04.770 ","End":"01:09.965","Text":"The image of f is the same as the domain of f to the minus 1 of x."},{"Start":"01:09.965 ","End":"01:13.415","Text":"Basically what we\u0027re asking for is,"},{"Start":"01:13.415 ","End":"01:19.600","Text":"what is the domain of f minus 1 of x,"},{"Start":"01:19.600 ","End":"01:22.110","Text":"which is the square root of x?"},{"Start":"01:22.110 ","End":"01:25.650","Text":"The answer to this is straightforward."},{"Start":"01:25.650 ","End":"01:27.530","Text":"x bigger or equal to 0."},{"Start":"01:27.530 ","End":"01:29.600","Text":"We\u0027ve seen this many times before."},{"Start":"01:29.600 ","End":"01:31.415","Text":"But in the case of image,"},{"Start":"01:31.415 ","End":"01:33.685","Text":"we write the letter y instead of x."},{"Start":"01:33.685 ","End":"01:38.365","Text":"The image is y bigger or equal to 0."},{"Start":"01:38.365 ","End":"01:44.405","Text":"The last part of the question is to show that f of f minus 1 of x is x."},{"Start":"01:44.405 ","End":"01:46.730","Text":"Let\u0027s see if we can compute this."},{"Start":"01:46.730 ","End":"01:53.210","Text":"f(f) to the minus 1 of x is equal to f of,"},{"Start":"01:53.210 ","End":"01:56.195","Text":"f minus 1 of x is the square root of x,"},{"Start":"01:56.195 ","End":"02:00.634","Text":"and f(x) is given to be x^2."},{"Start":"02:00.634 ","End":"02:06.665","Text":"This is the square root of x^2 and this is just equal to x,"},{"Start":"02:06.665 ","End":"02:08.660","Text":"which is what we needed to show."},{"Start":"02:08.660 ","End":"02:11.580","Text":"We finished all parts. We\u0027re done."}],"ID":4691},{"Watched":false,"Name":"Exercise 4 part a","Duration":"4m 38s","ChapterTopicVideoID":4680,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"In this exercise, which contains three parts,"},{"Start":"00:03.105 ","End":"00:06.000","Text":"A, B, and C, we\u0027re given a function f (x),"},{"Start":"00:06.000 ","End":"00:08.580","Text":"we have to find the inverse function f minus 1 of"},{"Start":"00:08.580 ","End":"00:11.820","Text":"x to identify the image of f. And finally,"},{"Start":"00:11.820 ","End":"00:15.030","Text":"as a check to show that f (f) minus 1 (x) is"},{"Start":"00:15.030 ","End":"00:18.660","Text":"x. I\u0027d like to mention that all these parts A, B,"},{"Start":"00:18.660 ","End":"00:22.665","Text":"and C appeared in a previous exercise where we showed that these functions"},{"Start":"00:22.665 ","End":"00:27.344","Text":"are indeed one-to-one and so the function does have an inverse."},{"Start":"00:27.344 ","End":"00:28.710","Text":"But in each of the cases,"},{"Start":"00:28.710 ","End":"00:30.810","Text":"let\u0027s just write instead of y,"},{"Start":"00:30.810 ","End":"00:33.460","Text":"the functional notation f (x)."},{"Start":"00:33.460 ","End":"00:36.740","Text":"So let\u0027s begin with part a."},{"Start":"00:36.740 ","End":"00:41.210","Text":"The usual technique with the inverse is to switch x with y."},{"Start":"00:41.210 ","End":"00:43.730","Text":"In fact, I\u0027ll just write that over here as"},{"Start":"00:43.730 ","End":"00:47.600","Text":"a reminder that we\u0027re going to use this three times."},{"Start":"00:47.600 ","End":"00:50.105","Text":"If y is twice x minus 3^2,"},{"Start":"00:50.105 ","End":"00:51.655","Text":"we write instead,"},{"Start":"00:51.655 ","End":"00:56.144","Text":"x equals twice y minus 3^2"},{"Start":"00:56.144 ","End":"01:02.780","Text":"minus 4 and also y is bigger or equal to 3."},{"Start":"01:02.780 ","End":"01:04.880","Text":"So switch x and y completely."},{"Start":"01:04.880 ","End":"01:07.370","Text":"Let\u0027s see if we can isolate y."},{"Start":"01:07.370 ","End":"01:12.230","Text":"So we get y minus 3^2."},{"Start":"01:12.230 ","End":"01:14.870","Text":"Assuming you\u0027re reasonably proficient with Algebra."},{"Start":"01:14.870 ","End":"01:17.690","Text":"So what I\u0027m going to do is take the 4 over to the other side and"},{"Start":"01:17.690 ","End":"01:20.780","Text":"divide by 2 that\u0027ll leave me that y minus"},{"Start":"01:20.780 ","End":"01:27.555","Text":"3^2 is equal to x plus 4 divided by 2."},{"Start":"01:27.555 ","End":"01:31.750","Text":"We have y minus 3^2 equals x plus 4/2."},{"Start":"01:31.750 ","End":"01:37.070","Text":"Normally, you would be tempted to write that y minus 3 is equal to"},{"Start":"01:37.070 ","End":"01:43.520","Text":"plus or minus square root of x plus 4/2."},{"Start":"01:43.520 ","End":"01:46.415","Text":"But we don\u0027t really need the plus or minus,"},{"Start":"01:46.415 ","End":"01:49.700","Text":"because if y is bigger or equal to 3,"},{"Start":"01:49.700 ","End":"01:54.815","Text":"this means that y minus 3 is bigger or equal to 0."},{"Start":"01:54.815 ","End":"01:58.970","Text":"We don\u0027t need the minus part and I\u0027m just going to erase it."},{"Start":"01:58.970 ","End":"02:00.890","Text":"So we just have the plus."},{"Start":"02:00.890 ","End":"02:03.365","Text":"Then adding 3 to both sides,"},{"Start":"02:03.365 ","End":"02:12.565","Text":"we get y=3 plus the square root of x plus 4/2."},{"Start":"02:12.565 ","End":"02:17.780","Text":"This is what we call f minus 1 of x."},{"Start":"02:17.780 ","End":"02:19.670","Text":"So that\u0027s that part."},{"Start":"02:19.670 ","End":"02:23.195","Text":"The next part is to find the image of the function."},{"Start":"02:23.195 ","End":"02:28.640","Text":"And the image of the function is the domain of the inverse function, which is this."},{"Start":"02:28.640 ","End":"02:29.975","Text":"So I have to ask,"},{"Start":"02:29.975 ","End":"02:34.355","Text":"what is the domain of f minus 1 of x?"},{"Start":"02:34.355 ","End":"02:37.195","Text":"What\u0027s the domain of all this part here?"},{"Start":"02:37.195 ","End":"02:39.690","Text":"Well, let\u0027s see, no problem with 3."},{"Start":"02:39.690 ","End":"02:43.970","Text":"The only problem would be a negative number under the square root sign."},{"Start":"02:43.970 ","End":"02:48.920","Text":"So all we have to require is what\u0027s under the square root sign be bigger or equal to 0."},{"Start":"02:48.920 ","End":"02:52.840","Text":"So we write x plus 4 over 2,"},{"Start":"02:52.840 ","End":"02:54.515","Text":"bigger or equal to 0,"},{"Start":"02:54.515 ","End":"02:57.350","Text":"multiply both sides by 2,"},{"Start":"02:57.350 ","End":"02:58.895","Text":"which is a positive number."},{"Start":"02:58.895 ","End":"03:01.910","Text":"So we don\u0027t have to change the sign of the inequality."},{"Start":"03:01.910 ","End":"03:05.074","Text":"So x plus 4 bigger or equal to 0,"},{"Start":"03:05.074 ","End":"03:09.260","Text":"and finally, x bigger or equal to minus 4."},{"Start":"03:09.260 ","End":"03:12.230","Text":"This is the domain of f minus 1 of x."},{"Start":"03:12.230 ","End":"03:14.930","Text":"But when we talk about the image of f,"},{"Start":"03:14.930 ","End":"03:16.880","Text":"we use y instead of x."},{"Start":"03:16.880 ","End":"03:22.840","Text":"So the answer for the image is y is bigger or equal to minus 4."},{"Start":"03:22.840 ","End":"03:25.565","Text":"So we found f minus 1, we found the image."},{"Start":"03:25.565 ","End":"03:33.155","Text":"The only thing that is left to do is to show that f(f) minus 1 of x is equal to x."},{"Start":"03:33.155 ","End":"03:36.470","Text":"Let\u0027s copy f (x) down below."},{"Start":"03:36.470 ","End":"03:42.295","Text":"We have to figure what is f (f) minus 1 of x."},{"Start":"03:42.295 ","End":"03:46.729","Text":"This equals f minus 1 of x we have over here."},{"Start":"03:46.729 ","End":"03:55.660","Text":"So it\u0027s f(3) plus the square root of x plus 4/2."},{"Start":"03:55.660 ","End":"03:59.205","Text":"This equals f(x) is given here,"},{"Start":"03:59.205 ","End":"04:02.760","Text":"so we have to put twice this whole thing,"},{"Start":"04:02.760 ","End":"04:12.195","Text":"3 plus the square root of x plus 4 over 2 minus 3^2 minus 4,"},{"Start":"04:12.195 ","End":"04:17.785","Text":"which equals, the 3 cancels with the minus 3."},{"Start":"04:17.785 ","End":"04:24.455","Text":"The square root squared is just the x plus 4 over 2 minus 4,"},{"Start":"04:24.455 ","End":"04:28.400","Text":"which equals 2 with the 2 cancels."},{"Start":"04:28.400 ","End":"04:29.960","Text":"And we have x plus 4,"},{"Start":"04:29.960 ","End":"04:32.255","Text":"4 with the 4 cancels."},{"Start":"04:32.255 ","End":"04:33.980","Text":"And we\u0027re left with just x."},{"Start":"04:33.980 ","End":"04:35.975","Text":"And that\u0027s what we had to show."},{"Start":"04:35.975 ","End":"04:38.910","Text":"We\u0027re done for part a."}],"ID":4689},{"Watched":false,"Name":"Exercise 4 part b","Duration":"7m 31s","ChapterTopicVideoID":4840,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.880","Text":"Now we come to part B."},{"Start":"00:03.980 ","End":"00:08.670","Text":"This is going to be my f(x) is what\u0027s on the right-hand side,"},{"Start":"00:08.670 ","End":"00:11.295","Text":"so this y is f(x)."},{"Start":"00:11.295 ","End":"00:16.980","Text":"Remember the technique is to switch x and y in order to find the inverse function."},{"Start":"00:16.980 ","End":"00:26.835","Text":"I\u0027m going to write this as x equals y squared minus 4y plus 5."},{"Start":"00:26.835 ","End":"00:32.410","Text":"The condition also gets replaced y is less than or equal to 2."},{"Start":"00:32.930 ","End":"00:35.625","Text":"The technique I\u0027m going to use here,"},{"Start":"00:35.625 ","End":"00:38.850","Text":"although I could use quadratic equations,"},{"Start":"00:38.850 ","End":"00:42.075","Text":"I\u0027m not going to, I\u0027m going to use completing the squares."},{"Start":"00:42.075 ","End":"00:45.740","Text":"Basically what completing the squares is just to remind you,"},{"Start":"00:45.740 ","End":"00:49.160","Text":"is writing the right-hand side of the perfect square."},{"Start":"00:49.160 ","End":"00:53.310","Text":"If it was y squared minus 4y plus 4,"},{"Start":"00:53.710 ","End":"00:56.030","Text":"it would be a perfect square."},{"Start":"00:56.030 ","End":"00:58.085","Text":"It would be y minus 2 all squared,"},{"Start":"00:58.085 ","End":"00:59.510","Text":"but I can\u0027t just change it,"},{"Start":"00:59.510 ","End":"01:03.305","Text":"so I\u0027ll put another plus 1 and now I\u0027m okay because 4 plus 1 is 5."},{"Start":"01:03.305 ","End":"01:09.330","Text":"This is equal to y minus 2 squared plus 1."},{"Start":"01:09.330 ","End":"01:11.400","Text":"Bring the 1 over to the left."},{"Start":"01:11.400 ","End":"01:16.540","Text":"X minus 1 equals y minus 2 squared."},{"Start":"01:18.340 ","End":"01:26.960","Text":"Then theoretically, we take the square root and say that the square root of x"},{"Start":"01:26.960 ","End":"01:36.095","Text":"minus 1 would be plus or minus y minus 2."},{"Start":"01:36.095 ","End":"01:38.420","Text":"Normally I would have the plus or minus,"},{"Start":"01:38.420 ","End":"01:42.460","Text":"but I claim that I can throughout one of them."},{"Start":"01:42.460 ","End":"01:47.930","Text":"I mean, if we look at this condition that y is less than or equal to 2,"},{"Start":"01:47.930 ","End":"01:52.475","Text":"isn\u0027t this just the same as y minus 2,"},{"Start":"01:52.475 ","End":"01:54.700","Text":"less than or equal to 0?"},{"Start":"01:54.700 ","End":"01:58.545","Text":"So y minus 2 has to be negative."},{"Start":"01:58.545 ","End":"02:02.480","Text":"If y minus 2 is negative and I want this square root to come up positive,"},{"Start":"02:02.480 ","End":"02:05.660","Text":"I have to take the minus because minus a negative will be positive."},{"Start":"02:05.660 ","End":"02:09.265","Text":"I\u0027m going to just erase the plus here."},{"Start":"02:09.265 ","End":"02:11.450","Text":"I\u0027m left with the minus,"},{"Start":"02:11.450 ","End":"02:17.310","Text":"and now I can just extract y. I\u0027ll continue over here and we get that y equals,"},{"Start":"02:17.310 ","End":"02:19.925","Text":"just bring it to this side and bring this to the other side."},{"Start":"02:19.925 ","End":"02:30.070","Text":"You can see that we get y equals 2 minus the square root of x minus 1."},{"Start":"02:30.710 ","End":"02:37.490","Text":"This expression here, this is my f minus"},{"Start":"02:37.490 ","End":"02:46.200","Text":"1 inverse of f(x)."},{"Start":"02:46.200 ","End":"02:48.795","Text":"Let\u0027s first of all identify"},{"Start":"02:48.795 ","End":"02:52.190","Text":"the image and then at the end we\u0027ll show that this thing holds."},{"Start":"02:52.190 ","End":"02:55.620","Text":"The image of f"},{"Start":"02:58.790 ","End":"03:03.230","Text":"is the domain of f minus 1."},{"Start":"03:03.230 ","End":"03:04.280","Text":"Now what\u0027s its domain?"},{"Start":"03:04.280 ","End":"03:05.870","Text":"Whereas this thing defined?"},{"Start":"03:05.870 ","End":"03:08.105","Text":"Well, the only problem is the square root."},{"Start":"03:08.105 ","End":"03:11.480","Text":"What\u0027s under the square root has to be bigger or equal to 0."},{"Start":"03:11.480 ","End":"03:16.805","Text":"This gives us that this x minus 1 has to be bigger or equal to 0,"},{"Start":"03:16.805 ","End":"03:21.515","Text":"which means that x is bigger or equal to 1."},{"Start":"03:21.515 ","End":"03:26.310","Text":"That\u0027s the domain of f minus 1,"},{"Start":"03:27.670 ","End":"03:31.460","Text":"which means that that is the image of"},{"Start":"03:31.460 ","End":"03:36.020","Text":"f. I almost forgot to say when we go to the image of f,"},{"Start":"03:36.020 ","End":"03:39.365","Text":"we now have to replace x back with y."},{"Start":"03:39.365 ","End":"03:43.325","Text":"In fact, y is bigger or equal to 1."},{"Start":"03:43.325 ","End":"03:49.430","Text":"That is the image of f. Image of f is y and domain of f is x,"},{"Start":"03:49.430 ","End":"03:53.100","Text":"and vice versa for the inverse function."},{"Start":"03:53.770 ","End":"04:01.974","Text":"Now let\u0027s show that f(f) minus 1(x) is x."},{"Start":"04:01.974 ","End":"04:08.685","Text":"What I\u0027m going to compute is f(f) minus 1(x)."},{"Start":"04:08.685 ","End":"04:12.920","Text":"Let\u0027s see what we get. Hopefully back to x again."},{"Start":"04:12.920 ","End":"04:15.425","Text":"This is equal to,"},{"Start":"04:15.425 ","End":"04:17.267","Text":"now f minus 1(x),"},{"Start":"04:17.267 ","End":"04:20.705","Text":"we\u0027ve already found this function is this,"},{"Start":"04:20.705 ","End":"04:28.440","Text":"is 2 minus the square root of x minus 1."},{"Start":"04:29.110 ","End":"04:32.130","Text":"That\u0027s the f minus 1(x),"},{"Start":"04:32.130 ","End":"04:33.300","Text":"but I need f of that."},{"Start":"04:33.300 ","End":"04:35.750","Text":"I still have an f around this."},{"Start":"04:35.750 ","End":"04:40.855","Text":"Now I apply the function f. This is our original function."},{"Start":"04:40.855 ","End":"04:46.350","Text":"It\u0027s this thing squared minus 4 times it plus 5."},{"Start":"04:46.350 ","End":"04:54.020","Text":"What I get is 2 minus the square root of x minus 1."},{"Start":"04:54.020 ","End":"05:03.410","Text":"All of this squared minus 4 times, let\u0027s see,"},{"Start":"05:03.410 ","End":"05:14.290","Text":"instead of x, we have 2 minus square root of x minus 1 and finally plus 5."},{"Start":"05:14.920 ","End":"05:19.580","Text":"Let\u0027s see now, raising this to the power of 2, squaring it,"},{"Start":"05:19.580 ","End":"05:24.905","Text":"it\u0027s one of these binomial squares."},{"Start":"05:24.905 ","End":"05:31.430","Text":"2 squared is 4 minus twice this times this"},{"Start":"05:31.430 ","End":"05:38.360","Text":"is minus 4 times the square root of x minus 1 plus the last one squared."},{"Start":"05:38.360 ","End":"05:40.130","Text":"When you square a square root,"},{"Start":"05:40.130 ","End":"05:42.510","Text":"you just get rid of the square root."},{"Start":"05:42.510 ","End":"05:46.435","Text":"But I\u0027ll leave it in brackets to show that this came from here."},{"Start":"05:46.435 ","End":"05:50.870","Text":"Now, minus 4 times 2 is 8, minus,"},{"Start":"05:50.870 ","End":"05:55.865","Text":"minus is plus square root of x minus 1,"},{"Start":"05:55.865 ","End":"05:58.850","Text":"and finally plus 5."},{"Start":"05:58.850 ","End":"06:02.820","Text":"Whoops! I just noticed I forgot the 4 here."},{"Start":"06:02.820 ","End":"06:06.470","Text":"Now we can cancel some things."},{"Start":"06:06.470 ","End":"06:11.330","Text":"The main thing that I\u0027m happy about is that the square roots disappear because look,"},{"Start":"06:11.330 ","End":"06:16.925","Text":"we have a minus 4 times the square root and a plus 4 times the square root."},{"Start":"06:16.925 ","End":"06:20.720","Text":"Except for that, we look at the numbers,"},{"Start":"06:20.720 ","End":"06:24.485","Text":"we have 4 minus 1,"},{"Start":"06:24.485 ","End":"06:28.015","Text":"I\u0027ll just mark them as 4 minus 1,"},{"Start":"06:28.015 ","End":"06:30.720","Text":"minus 8 plus 5."},{"Start":"06:30.720 ","End":"06:31.980","Text":"The pluses are 9,"},{"Start":"06:31.980 ","End":"06:33.405","Text":"the minuses are 9."},{"Start":"06:33.405 ","End":"06:36.225","Text":"The numbers disappear also."},{"Start":"06:36.225 ","End":"06:40.140","Text":"In fact, all that we\u0027re left with is x."},{"Start":"06:40.140 ","End":"06:41.995","Text":"This is what we wanted to show,"},{"Start":"06:41.995 ","End":"06:46.300","Text":"or as we say in Latin, quod erat demonstrandum."},{"Start":"06:46.300 ","End":"06:50.510","Text":"We\u0027ve answered all the questions."},{"Start":"06:51.650 ","End":"06:58.375","Text":"We\u0027ve shown the founding the inverse function,"},{"Start":"06:58.375 ","End":"07:00.070","Text":"and that is here."},{"Start":"07:00.070 ","End":"07:03.530","Text":"Let me highlight that."},{"Start":"07:05.660 ","End":"07:08.475","Text":"That\u0027s the inverse function."},{"Start":"07:08.475 ","End":"07:15.658","Text":"Let\u0027s see we have to find the image and that is this."},{"Start":"07:15.658 ","End":"07:22.535","Text":"The proof is the proof, can\u0027t highlight that."},{"Start":"07:22.535 ","End":"07:26.810","Text":"But anyway, what I can highlight is that I started from this,"},{"Start":"07:26.810 ","End":"07:29.315","Text":"and I ended up with this."},{"Start":"07:29.315 ","End":"07:32.730","Text":"We\u0027re done with this part."}],"ID":4840},{"Watched":false,"Name":"Exercise 4 part c","Duration":"5m 55s","ChapterTopicVideoID":4681,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.584","Text":"Finally, part C, the usual technique,"},{"Start":"00:03.584 ","End":"00:08.560","Text":"we switch x with y here and we get x=y^2"},{"Start":"00:08.780 ","End":"00:16.590","Text":"over y^2 plus 1 y bigger or equal to 0."},{"Start":"00:16.590 ","End":"00:19.350","Text":"We want to try and isolate y in terms of x."},{"Start":"00:19.350 ","End":"00:22.720","Text":"Let\u0027s multiply both sides by the denominator."},{"Start":"00:22.720 ","End":"00:27.940","Text":"Xy^2 plus x=y^2."},{"Start":"00:27.940 ","End":"00:31.725","Text":"Let\u0027s bring it over minus y^2=0."},{"Start":"00:31.725 ","End":"00:37.710","Text":"Then let\u0027s take y^2 outside the brackets from the first and last term."},{"Start":"00:37.710 ","End":"00:41.110","Text":"We get x minus 1,"},{"Start":"00:41.110 ","End":"00:44.465","Text":"y^2 plus x=0,"},{"Start":"00:44.465 ","End":"00:48.995","Text":"and then throwing x to the other side and dividing by the coefficient of y^2,"},{"Start":"00:48.995 ","End":"00:56.480","Text":"we get y^2 equals minus x over x minus 1."},{"Start":"00:56.480 ","End":"00:58.355","Text":"Taking the square root,"},{"Start":"00:58.355 ","End":"01:05.840","Text":"we get y equals the square root of minus x over x minus 1."},{"Start":"01:05.840 ","End":"01:10.015","Text":"You might wonder, why shouldn\u0027t we write plus or minus here?"},{"Start":"01:10.015 ","End":"01:13.480","Text":"The answer is because y is bigger or equal to 0,"},{"Start":"01:13.480 ","End":"01:16.610","Text":"so I\u0027m going to erase that plus and minus."},{"Start":"01:16.610 ","End":"01:20.640","Text":"This is the inverse function,"},{"Start":"01:21.880 ","End":"01:24.370","Text":"f^-1(x). That\u0027s that part."},{"Start":"01:24.370 ","End":"01:28.595","Text":"Next we have to find the image of f(x),"},{"Start":"01:28.595 ","End":"01:34.550","Text":"which means that we have to find the domain of the inverse function."},{"Start":"01:34.550 ","End":"01:37.024","Text":"We have here a square root function,"},{"Start":"01:37.024 ","End":"01:39.410","Text":"and there\u0027s also a fraction, the domain,"},{"Start":"01:39.410 ","End":"01:42.305","Text":"which means all the values of x we can substitute."},{"Start":"01:42.305 ","End":"01:44.900","Text":"We have to take into account 2 things."},{"Start":"01:44.900 ","End":"01:47.705","Text":"This denominator must not be 0."},{"Start":"01:47.705 ","End":"01:51.650","Text":"In other words, x minus 1 cannot be 0."},{"Start":"01:51.650 ","End":"01:57.190","Text":"The other thing is that whatever comes under the square root sign has to be non-negative."},{"Start":"01:57.190 ","End":"02:04.280","Text":"Minus x over x minus 1 has to be bigger or equal to 0."},{"Start":"02:04.280 ","End":"02:06.890","Text":"We could hear just quickly finish off by"},{"Start":"02:06.890 ","End":"02:10.315","Text":"saying that X must not equal to 1. Back to the second bit."},{"Start":"02:10.315 ","End":"02:15.860","Text":"We can\u0027t simply multiply by x minus 1 because we have here an inequality."},{"Start":"02:15.860 ","End":"02:22.730","Text":"1 of the tricks that I use is to multiply both sides by (x-1)^2,"},{"Start":"02:22.730 ","End":"02:24.970","Text":"which is not going to be negative."},{"Start":"02:24.970 ","End":"02:26.600","Text":"If we do that,"},{"Start":"02:26.600 ","End":"02:30.995","Text":"we get minus x times x minus 1."},{"Start":"02:30.995 ","End":"02:39.510","Text":"Because 1 of the x minus 1 gets swallowed by the denominator is bigger or equal to 0."},{"Start":"02:39.510 ","End":"02:42.485","Text":"If I multiply by minus 1,"},{"Start":"02:42.485 ","End":"02:44.360","Text":"I\u0027d rather have a plus here."},{"Start":"02:44.360 ","End":"02:46.430","Text":"Multiplying by minus 1,"},{"Start":"02:46.430 ","End":"02:49.400","Text":"I would get less than or equal to 0,"},{"Start":"02:49.400 ","End":"02:54.130","Text":"and I would get here x times x minus 1."},{"Start":"02:54.130 ","End":"02:56.105","Text":"Now, this is a quadratic function."},{"Start":"02:56.105 ","End":"02:58.565","Text":"I could expand it to x^2 minus x."},{"Start":"02:58.565 ","End":"03:02.255","Text":"It\u0027s a quadratic function where a=1,"},{"Start":"03:02.255 ","End":"03:05.330","Text":"which means that we have an upward-facing parabola."},{"Start":"03:05.330 ","End":"03:07.840","Text":"Let\u0027s make a quick sketch of this here."},{"Start":"03:07.840 ","End":"03:10.590","Text":"The roots are clearly 0 and 1,"},{"Start":"03:10.590 ","End":"03:15.015","Text":"I can mark here 0 and mark here 1."},{"Start":"03:15.015 ","End":"03:18.140","Text":"The parabola is something like this."},{"Start":"03:18.140 ","End":"03:22.610","Text":"In any case, what we want is less than or equal to 0,"},{"Start":"03:22.610 ","End":"03:27.159","Text":"which means that we want this bit here,"},{"Start":"03:27.159 ","End":"03:31.445","Text":"but less than or equal to means we\u0027re including these 2 points."},{"Start":"03:31.445 ","End":"03:35.795","Text":"What we see is that this occurs simply in this region here."},{"Start":"03:35.795 ","End":"03:40.565","Text":"In other words, zero less than or equal to x,"},{"Start":"03:40.565 ","End":"03:42.995","Text":"less than or equal to 1,"},{"Start":"03:42.995 ","End":"03:46.400","Text":"but remember that x cannot equal 1."},{"Start":"03:46.400 ","End":"03:49.280","Text":"If we combine this with this,"},{"Start":"03:49.280 ","End":"03:54.065","Text":"we would get zero less than or equal to x"},{"Start":"03:54.065 ","End":"04:00.020","Text":"less than 1.The final thing to remember is replace x with y."},{"Start":"04:00.020 ","End":"04:01.400","Text":"This is the domain of"},{"Start":"04:01.400 ","End":"04:06.170","Text":"the inverse function but the image of the function is the same thing just with y,"},{"Start":"04:06.170 ","End":"04:09.995","Text":"because y is a typical variable in the image."},{"Start":"04:09.995 ","End":"04:13.595","Text":"This is the answer to the image."},{"Start":"04:13.595 ","End":"04:15.275","Text":"We\u0027re still not done."},{"Start":"04:15.275 ","End":"04:21.979","Text":"We still have to show that f(f^-1(x))= x identically."},{"Start":"04:21.979 ","End":"04:25.100","Text":"Let\u0027s just start working on it and see if we get x in the end."},{"Start":"04:25.100 ","End":"04:30.670","Text":"This is equal to f(x) was x^2 over x^2 plus 1."},{"Start":"04:30.670 ","End":"04:33.200","Text":"Now, f^-1(x),"},{"Start":"04:33.200 ","End":"04:36.395","Text":"we already figured is this expression with the square root."},{"Start":"04:36.395 ","End":"04:44.360","Text":"This is f of square root of minus x over x minus 1."},{"Start":"04:44.360 ","End":"04:49.040","Text":"This equals, now replacing x with this whole thing."},{"Start":"04:49.040 ","End":"04:50.510","Text":"Wherever we see x^2,"},{"Start":"04:50.510 ","End":"04:52.535","Text":"we just remove the square root sign."},{"Start":"04:52.535 ","End":"04:58.100","Text":"Basically, what we get is minus x over x minus 1,"},{"Start":"04:58.100 ","End":"05:02.105","Text":"which is the square root ^2 over the same thing,"},{"Start":"05:02.105 ","End":"05:07.205","Text":"minus x over x minus 1 plus 1."},{"Start":"05:07.205 ","End":"05:09.850","Text":"Need to do a little bit of algebra here."},{"Start":"05:09.850 ","End":"05:12.500","Text":"We have a fraction with fractions in"},{"Start":"05:12.500 ","End":"05:16.355","Text":"both numerator and denominator but if instead of this one,"},{"Start":"05:16.355 ","End":"05:20.800","Text":"I write it as x minus 1 over x minus 1."},{"Start":"05:20.800 ","End":"05:24.050","Text":"I can multiply top and bottom by x minus 1."},{"Start":"05:24.050 ","End":"05:27.260","Text":"Basically, I can cancel all the x minus 1s."},{"Start":"05:27.260 ","End":"05:30.500","Text":"In other words, I\u0027m canceling this, this and this."},{"Start":"05:30.500 ","End":"05:39.500","Text":"That gives me minus x over minus x plus x minus 1 from here."},{"Start":"05:39.500 ","End":"05:41.300","Text":"Minus x and x cancel,"},{"Start":"05:41.300 ","End":"05:46.490","Text":"so all I\u0027m left with is minus x over minus 1."},{"Start":"05:46.490 ","End":"05:49.865","Text":"This is exactly equal to just x,"},{"Start":"05:49.865 ","End":"05:52.240","Text":"which is what we wanted to show."},{"Start":"05:52.240 ","End":"05:56.200","Text":"We\u0027re done with part c and the whole exercise."}],"ID":4690},{"Watched":false,"Name":"Exercise 5","Duration":"12m 34s","ChapterTopicVideoID":4841,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Here we have 3 exercises, really a, b,"},{"Start":"00:03.330 ","End":"00:05.040","Text":"and c. In each one,"},{"Start":"00:05.040 ","End":"00:06.510","Text":"we\u0027re given f(x),"},{"Start":"00:06.510 ","End":"00:12.970","Text":"we have to find the inverse function f^-1 and also the image of the function f,"},{"Start":"00:12.970 ","End":"00:16.935","Text":"and finally, to show that this equality holds."},{"Start":"00:16.935 ","End":"00:20.320","Text":"Let\u0027s begin with a."},{"Start":"00:22.280 ","End":"00:25.725","Text":"The usual technique is to"},{"Start":"00:25.725 ","End":"00:33.600","Text":"switch y with x and x with y and that helps us to find the inverse."},{"Start":"00:33.600 ","End":"00:37.350","Text":"We\u0027ll rewrite a as,"},{"Start":"00:37.350 ","End":"00:39.855","Text":"instead of y=4 natural log of x."},{"Start":"00:39.855 ","End":"00:43.620","Text":"We\u0027ll write x=4 natural log of"},{"Start":"00:43.620 ","End":"00:51.545","Text":"y and we have to extract y in terms of x and this trick will give us the inverse."},{"Start":"00:51.545 ","End":"00:56.825","Text":"Let\u0027s see. We\u0027ll divide both sides by 4,"},{"Start":"00:56.825 ","End":"01:02.100","Text":"x/4=natural log of y."},{"Start":"01:03.730 ","End":"01:09.305","Text":"We raise each side as an exponent,"},{"Start":"01:09.305 ","End":"01:14.050","Text":"e to the power of each side so we get e^x/4."},{"Start":"01:14.050 ","End":"01:15.640","Text":"Let\u0027s also switch sides."},{"Start":"01:15.640 ","End":"01:21.160","Text":"So e^x/4=e^natural log,"},{"Start":"01:21.160 ","End":"01:25.900","Text":"the e and the natural log cancel each other out and we get just y."},{"Start":"01:25.900 ","End":"01:33.390","Text":"Or in general, if a=natural log of b,"},{"Start":"01:33.390 ","End":"01:37.085","Text":"that\u0027s the same thing as b= e^a."},{"Start":"01:37.085 ","End":"01:40.910","Text":"That\u0027s the definition really of the natural log."},{"Start":"01:41.510 ","End":"01:46.380","Text":"I\u0027ll just dispense with that. Let\u0027s continue."},{"Start":"01:46.380 ","End":"01:55.980","Text":"This will be our inverse function and I will write it"},{"Start":"01:55.980 ","End":"02:05.085","Text":"as f ^-1(x) equals e^x/4."},{"Start":"02:05.085 ","End":"02:10.730","Text":"That\u0027s one part of the question, so highlight that."},{"Start":"02:10.730 ","End":"02:13.270","Text":"To find the image of f,"},{"Start":"02:13.270 ","End":"02:17.335","Text":"we need to look at the domain of f^-1."},{"Start":"02:17.335 ","End":"02:26.465","Text":"Let\u0027s see what is the domain of f^-1."},{"Start":"02:26.465 ","End":"02:30.685","Text":"Just looking at this, this thing is defined for all x,"},{"Start":"02:30.685 ","End":"02:35.155","Text":"so we just write all x."},{"Start":"02:35.155 ","End":"02:42.690","Text":"That\u0027s the same as the image of f,"},{"Start":"02:42.690 ","End":"02:44.820","Text":"except that in this case,"},{"Start":"02:44.820 ","End":"02:47.760","Text":"we revert back to y,"},{"Start":"02:47.760 ","End":"02:50.385","Text":"so this is all y."},{"Start":"02:50.385 ","End":"02:53.685","Text":"That\u0027s another part of the question."},{"Start":"02:53.685 ","End":"02:56.535","Text":"I\u0027ll highlight that also."},{"Start":"02:56.535 ","End":"03:04.215","Text":"Now the final part is to show this equality and I\u0027ll do it over here."},{"Start":"03:04.215 ","End":"03:11.015","Text":"Let\u0027s see, f(f^-1(x))."},{"Start":"03:11.015 ","End":"03:15.080","Text":"Let\u0027s do a series of steps and hope we get to x in the end."},{"Start":"03:15.080 ","End":"03:16.805","Text":"This is f of,"},{"Start":"03:16.805 ","End":"03:22.400","Text":"now f^-1(x) is e^x/4."},{"Start":"03:22.400 ","End":"03:25.670","Text":"This is equal to, using the definition of f,"},{"Start":"03:25.670 ","End":"03:28.730","Text":"which is 4 natural log of x,"},{"Start":"03:28.730 ","End":"03:35.335","Text":"is equal to 4 natural log of this thing of e^x/4,"},{"Start":"03:35.335 ","End":"03:40.115","Text":"which equals the natural log and the exponent cancel each other out,"},{"Start":"03:40.115 ","End":"03:45.370","Text":"so it\u0027s just x/4, which equals x."},{"Start":"03:45.370 ","End":"03:46.920","Text":"That answers the last part,"},{"Start":"03:46.920 ","End":"03:49.360","Text":"so we\u0027re done with a."},{"Start":"03:49.850 ","End":"03:55.860","Text":"Now, let\u0027s go on to part b and in b,"},{"Start":"03:55.860 ","End":"03:59.475","Text":"same as in a using the same techniques,"},{"Start":"03:59.475 ","End":"04:07.260","Text":"we\u0027ll replace y with x and x with y and this will help us find the inverse function."},{"Start":"04:07.260 ","End":"04:10.740","Text":"Instead of y equals this thing with x,"},{"Start":"04:10.740 ","End":"04:19.380","Text":"I\u0027ll write x=2 plus 3 natural log of y minus 1."},{"Start":"04:19.380 ","End":"04:23.280","Text":"The idea is to extract y in terms of x."},{"Start":"04:23.800 ","End":"04:31.070","Text":"Let\u0027s bring the 2 to the other side and switch sides."},{"Start":"04:31.070 ","End":"04:40.295","Text":"I\u0027ve got 3 natural log of y minus 1 equals x minus 2."},{"Start":"04:40.295 ","End":"04:49.380","Text":"Divide by 3 natural log of y minus 1 equals x minus 2/3."},{"Start":"04:52.040 ","End":"04:57.760","Text":"Then if this is the natural log of this and this is the exponent of this,"},{"Start":"04:57.760 ","End":"05:05.720","Text":"so y minus 1= e^x minus 2 over 3."},{"Start":"05:05.720 ","End":"05:09.995","Text":"Finally, we get y equals"},{"Start":"05:09.995 ","End":"05:19.970","Text":"1 plus e^x minus 2 over 3 and this will be our f^-1 (x),"},{"Start":"05:19.970 ","End":"05:22.685","Text":"which I shall highlight."},{"Start":"05:22.685 ","End":"05:27.715","Text":"There we go. Now we have to identify the image."},{"Start":"05:27.715 ","End":"05:29.590","Text":"To find the image of f,"},{"Start":"05:29.590 ","End":"05:33.870","Text":"we ask what is the domain of the inverse function."},{"Start":"05:33.870 ","End":"05:37.555","Text":"The domain of f^-1. Well, let\u0027s look at it."},{"Start":"05:37.555 ","End":"05:44.275","Text":"What could possibly be a bad value of x to try and substitute?"},{"Start":"05:44.275 ","End":"05:46.540","Text":"For any x, we could subtract 2."},{"Start":"05:46.540 ","End":"05:49.090","Text":"Any number can be divided by 3."},{"Start":"05:49.090 ","End":"05:52.585","Text":"Any number we can raise e to the power of. We can always add 1."},{"Start":"05:52.585 ","End":"05:55.030","Text":"Any x can be substituted here,"},{"Start":"05:55.030 ","End":"05:59.500","Text":"so the answer to this one is all x."},{"Start":"05:59.500 ","End":"06:03.620","Text":"When we take the image of f,"},{"Start":"06:06.020 ","End":"06:11.930","Text":"it will be not all x and we have to replace it back with y."},{"Start":"06:12.180 ","End":"06:15.505","Text":"That answers the next bit."},{"Start":"06:15.505 ","End":"06:25.165","Text":"Finally, we want to show that equality holds this identity."},{"Start":"06:25.165 ","End":"06:28.900","Text":"See if I have room here, go for it."},{"Start":"06:28.900 ","End":"06:38.090","Text":"f(f^-1), which is the inverse of f(x) is equal to."},{"Start":"06:38.090 ","End":"06:40.275","Text":"I work from the inside out."},{"Start":"06:40.275 ","End":"06:44.590","Text":"First of all, is the inverse which I take from here,"},{"Start":"06:44.590 ","End":"06:53.675","Text":"so this is f(1 plus e^x minus 2 over 3),"},{"Start":"06:53.675 ","End":"07:00.930","Text":"which equals, now applying f and f is what\u0027s written here."},{"Start":"07:00.930 ","End":"07:07.095","Text":"It\u0027s 2 plus 3 natural log."},{"Start":"07:07.095 ","End":"07:14.040","Text":"Instead of x, I need to put this whole expression so it\u0027s"},{"Start":"07:14.040 ","End":"07:22.800","Text":"1 plus e^x minus"},{"Start":"07:22.800 ","End":"07:24.685","Text":"2 over 3."},{"Start":"07:24.685 ","End":"07:31.885","Text":"I\u0027ll put it in brackets for emphasis that this takes the place of x here, minus 1."},{"Start":"07:31.885 ","End":"07:35.695","Text":"It looks a mess, it\u0027ll soon straighten itself out."},{"Start":"07:35.695 ","End":"07:40.940","Text":"This is equal to 2 plus 3 natural log."},{"Start":"07:40.940 ","End":"07:45.180","Text":"Now, the one with the minus 1 cancel,"},{"Start":"07:45.180 ","End":"07:52.600","Text":"so all I\u0027m left with is e^x minus 2 over 3."},{"Start":"07:52.670 ","End":"07:58.375","Text":"This is equal to natural log of an exponent is just what\u0027s here."},{"Start":"07:58.375 ","End":"08:04.180","Text":"It\u0027s 2 plus 3 times x minus 2 over"},{"Start":"08:04.180 ","End":"08:13.230","Text":"3 and this equals 3 times 3, this cancels out."},{"Start":"08:13.230 ","End":"08:15.360","Text":"Running out of space here."},{"Start":"08:15.360 ","End":"08:18.330","Text":"Just 2 plus 3s cancel,"},{"Start":"08:18.330 ","End":"08:23.620","Text":"so it\u0027s x minus 2, which equals x."},{"Start":"08:24.440 ","End":"08:27.140","Text":"We\u0027ve shown this because you see,"},{"Start":"08:27.140 ","End":"08:31.370","Text":"we started with this and we ended with this and that\u0027s what we had to do."},{"Start":"08:31.370 ","End":"08:33.875","Text":"Onto the next question."},{"Start":"08:33.875 ","End":"08:39.385","Text":"Now we\u0027ve reached part C, the same technique."},{"Start":"08:39.385 ","End":"08:42.495","Text":"Just go with a quicker pace now."},{"Start":"08:42.495 ","End":"08:45.080","Text":"We take this y equals this,"},{"Start":"08:45.080 ","End":"08:51.830","Text":"and replace x with y. X equals 1 plus 2 e^2y."},{"Start":"08:51.830 ","End":"08:57.040","Text":"Now try and isolate y. Subtract 1, x minus 1."},{"Start":"08:57.040 ","End":"09:02.600","Text":"You know what? We\u0027ll divide by 2 all in one go, equals e^2y."},{"Start":"09:02.600 ","End":"09:05.950","Text":"Take the natural log of"},{"Start":"09:15.560 ","End":"09:21.835","Text":"x minus 1 over 2 equals 2y."},{"Start":"09:21.835 ","End":"09:25.294","Text":"Then I\u0027ll switch sides also."},{"Start":"09:25.294 ","End":"09:31.190","Text":"y equals 1/2 natural log of x minus"},{"Start":"09:31.190 ","End":"09:38.020","Text":"1 over 2 and this will be my f^-1(x)."},{"Start":"09:41.420 ","End":"09:45.390","Text":"That\u0027s the first bit we found f^-1(x)."},{"Start":"09:45.390 ","End":"09:47.730","Text":"Now, the image."},{"Start":"09:47.730 ","End":"09:49.020","Text":"For the image of f,"},{"Start":"09:49.020 ","End":"09:52.320","Text":"we need the domain of f^-1."},{"Start":"09:52.320 ","End":"09:58.780","Text":"The inverse, the domain of f^-1."},{"Start":"09:59.540 ","End":"10:02.045","Text":"Let\u0027s see what could that be."},{"Start":"10:02.045 ","End":"10:06.820","Text":"In other words, what are the bad values of x or the good values of x?"},{"Start":"10:06.820 ","End":"10:14.670","Text":"The only thing to worry about is that the natural log has to have a positive argument."},{"Start":"10:14.670 ","End":"10:16.725","Text":"Just do this at the sides."},{"Start":"10:16.725 ","End":"10:23.625","Text":"What we have is x minus 1 over 2 has got to be strictly positive,"},{"Start":"10:23.625 ","End":"10:26.760","Text":"which means that x minus 1 has to also"},{"Start":"10:26.760 ","End":"10:29.990","Text":"be positive because if something over 2 is positive,"},{"Start":"10:29.990 ","End":"10:31.474","Text":"it has to be positive."},{"Start":"10:31.474 ","End":"10:33.890","Text":"x has to be bigger than 1."},{"Start":"10:33.890 ","End":"10:37.330","Text":"The domain is x bigger than 1."},{"Start":"10:37.330 ","End":"10:43.790","Text":"This means that the image of f(x) is the same thing,"},{"Start":"10:43.790 ","End":"10:46.880","Text":"but we have to revert back to y."},{"Start":"10:46.880 ","End":"10:50.940","Text":"Is y bigger than 1?"},{"Start":"10:52.550 ","End":"10:56.495","Text":"That answers that part."},{"Start":"10:56.495 ","End":"10:59.540","Text":"Now finally, we have to show this."},{"Start":"10:59.540 ","End":"11:05.320","Text":"I\u0027ll start with the left-hand side and see if I can get to the right-hand side."},{"Start":"11:05.320 ","End":"11:11.515","Text":"f(f^-1(x))."},{"Start":"11:11.515 ","End":"11:13.550","Text":"Let\u0027s see what this equals."},{"Start":"11:13.550 ","End":"11:16.190","Text":"This equals f of,"},{"Start":"11:16.190 ","End":"11:25.390","Text":"f^-1(x) is written here 1/2 natural log of x minus 1 over 2."},{"Start":"11:26.690 ","End":"11:31.905","Text":"We have the formula for f. It\u0027s this 1 plus 2e^2x."},{"Start":"11:31.905 ","End":"11:39.660","Text":"This equals 1 plus 2e^2 and instead of x,"},{"Start":"11:39.660 ","End":"11:41.565","Text":"I\u0027ll put this whole thing here,"},{"Start":"11:41.565 ","End":"11:50.605","Text":"so it\u0027s 2 times 1/2 times natural log of x minus 1 over 2."},{"Start":"11:50.605 ","End":"11:53.360","Text":"Let\u0027s see how we can simplify this."},{"Start":"11:53.360 ","End":"12:00.125","Text":"This equals 1 plus 2e^2 with the 1/2 cancels."},{"Start":"12:00.125 ","End":"12:05.165","Text":"That\u0027s just the natural log of x minus 1 over 2."},{"Start":"12:05.165 ","End":"12:06.680","Text":"We\u0027ve seen this before."},{"Start":"12:06.680 ","End":"12:09.470","Text":"When you take e to the power of natural log,"},{"Start":"12:09.470 ","End":"12:11.435","Text":"they cancel each other out."},{"Start":"12:11.435 ","End":"12:15.990","Text":"We\u0027re just left with x minus 1 over 2."},{"Start":"12:16.370 ","End":"12:18.630","Text":"Now, this is equal to,"},{"Start":"12:18.630 ","End":"12:20.220","Text":"the 2 with the 2 cancels,"},{"Start":"12:20.220 ","End":"12:25.365","Text":"it\u0027s 1 plus x minus 1 and this is equal to x."},{"Start":"12:25.365 ","End":"12:29.615","Text":"You see we started off with this and ended up with this,"},{"Start":"12:29.615 ","End":"12:33.840","Text":"which proves this, and so we\u0027re done."}],"ID":4841},{"Watched":false,"Name":"Exercise 6","Duration":"2m 29s","ChapterTopicVideoID":7793,"CourseChapterTopicPlaylistID":1185,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we have to find the inverse function of this function f(x),"},{"Start":"00:04.770 ","End":"00:07.665","Text":"which is log to the base 2 of this expression,"},{"Start":"00:07.665 ","End":"00:10.380","Text":"the domain is x bigger than 1,"},{"Start":"00:10.380 ","End":"00:12.540","Text":"and when x is bigger than 1,"},{"Start":"00:12.540 ","End":"00:14.040","Text":"there\u0027s certainly not 0,"},{"Start":"00:14.040 ","End":"00:15.960","Text":"so 1 over x is okay."},{"Start":"00:15.960 ","End":"00:18.540","Text":"Also this thing, it will be positive."},{"Start":"00:18.540 ","End":"00:20.850","Text":"I could have even written x is bigger than 0 and it"},{"Start":"00:20.850 ","End":"00:23.475","Text":"would still be positive and everything would be okay."},{"Start":"00:23.475 ","End":"00:25.830","Text":"Anyway, let\u0027s let y equal f(x),"},{"Start":"00:25.830 ","End":"00:27.705","Text":"so y is equal to this."},{"Start":"00:27.705 ","End":"00:30.870","Text":"Now, to find the inverse function is 1 way of doing it"},{"Start":"00:30.870 ","End":"00:34.210","Text":"is to get x in terms of y step-by-step."},{"Start":"00:34.210 ","End":"00:36.470","Text":"Let\u0027s first of all get rid of the log to the base 2,"},{"Start":"00:36.470 ","End":"00:39.430","Text":"and this is what we get just using the rules of logarithms."},{"Start":"00:39.430 ","End":"00:41.320","Text":"Log to the base 2 of this is this,"},{"Start":"00:41.320 ","End":"00:43.470","Text":"then 2 to the this is this anyway."},{"Start":"00:43.470 ","End":"00:45.830","Text":"Now we want to get rid of the denominator here,"},{"Start":"00:45.830 ","End":"00:48.470","Text":"multiply everything by x, and I get this."},{"Start":"00:48.470 ","End":"00:51.635","Text":"You can see it\u0027s a quadratic equation in x,"},{"Start":"00:51.635 ","End":"00:53.950","Text":"especially when I rewrite it like this."},{"Start":"00:53.950 ","End":"00:58.260","Text":"I\u0027m going to solve this quadratic equation using the quadratic formula,"},{"Start":"00:58.260 ","End":"00:59.990","Text":"and this is what the formula gives."},{"Start":"00:59.990 ","End":"01:03.110","Text":"The trouble is that there\u0027s x_1 and x_2 and we can"},{"Start":"01:03.110 ","End":"01:06.515","Text":"take the plus and we can take the minus and that wouldn\u0027t be a function."},{"Start":"01:06.515 ","End":"01:09.650","Text":"However, because x is bigger than 1,"},{"Start":"01:09.650 ","End":"01:12.200","Text":"it turns out that only the plus is a solution."},{"Start":"01:12.200 ","End":"01:14.240","Text":"I\u0027m not going to go into all the algebra,"},{"Start":"01:14.240 ","End":"01:15.320","Text":"but if I take the minus,"},{"Start":"01:15.320 ","End":"01:16.910","Text":"it\u0027s not going to be bigger than 1,"},{"Start":"01:16.910 ","End":"01:19.760","Text":"and so we now have x in terms of y."},{"Start":"01:19.760 ","End":"01:22.430","Text":"Now, this actually gives us the inverse function,"},{"Start":"01:22.430 ","End":"01:23.930","Text":"but not quite in the way we want."},{"Start":"01:23.930 ","End":"01:27.695","Text":"It gives us x is the inverse function of y,"},{"Start":"01:27.695 ","End":"01:31.595","Text":"but we want to write the inverse function as y in terms of x,"},{"Start":"01:31.595 ","End":"01:33.065","Text":"the letters are dummy variables."},{"Start":"01:33.065 ","End":"01:35.119","Text":"I just switch the x and the y."},{"Start":"01:35.119 ","End":"01:36.365","Text":"Perhaps I\u0027ll write that here."},{"Start":"01:36.365 ","End":"01:38.690","Text":"We switched x and y to get"},{"Start":"01:38.690 ","End":"01:42.230","Text":"the function because the function doesn\u0027t depend on the variable."},{"Start":"01:42.230 ","End":"01:43.460","Text":"This is x as a function of y,"},{"Start":"01:43.460 ","End":"01:44.990","Text":"so this is the function of x."},{"Start":"01:44.990 ","End":"01:46.790","Text":"You replace y by x."},{"Start":"01:46.790 ","End":"01:51.785","Text":"The last thing I want us to relate to the domain of the function,"},{"Start":"01:51.785 ","End":"01:54.560","Text":"when is this f to the minus 1 defined?"},{"Start":"01:54.560 ","End":"01:58.835","Text":"Well, what\u0027s under the square root sign has to be non-negative."},{"Start":"01:58.835 ","End":"02:03.005","Text":"This what\u0027s under the square root has to be bigger or equal to 0 for the domain."},{"Start":"02:03.005 ","End":"02:04.400","Text":"This is not hard to solve."},{"Start":"02:04.400 ","End":"02:08.870","Text":"Just bring the 4 to the other side and notice that 4 is 2^2."},{"Start":"02:08.870 ","End":"02:11.540","Text":"Now I have 2 things that equal with equal basis."},{"Start":"02:11.540 ","End":"02:13.575","Text":"There\u0027s a 2 here and a 2 here,"},{"Start":"02:13.575 ","End":"02:16.520","Text":"and so since the base is bigger than 1,"},{"Start":"02:16.520 ","End":"02:18.005","Text":"it\u0027s an increasing function,"},{"Start":"02:18.005 ","End":"02:20.270","Text":"2x is bigger than 2,"},{"Start":"02:20.270 ","End":"02:22.730","Text":"and finally x bigger or equal to 1."},{"Start":"02:22.730 ","End":"02:29.910","Text":"To summarize, we found the inverse function here and its domain is here. We\u0027re done."}],"ID":7866}],"Thumbnail":null,"ID":1185},{"Name":"Piecewise-Defined Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Piecewise-Defined Functions","Duration":"17m 57s","ChapterTopicVideoID":1218,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this clip, we\u0027ll be learning about"},{"Start":"00:02.565 ","End":"00:07.260","Text":"piecewise-defined functions and they also have some other names I\u0027ll get to that."},{"Start":"00:07.260 ","End":"00:10.845","Text":"I\u0027ll start straight away with an example."},{"Start":"00:10.845 ","End":"00:15.270","Text":"An example where we take 2 different functions and somehow"},{"Start":"00:15.270 ","End":"00:19.725","Text":"build a third one out of them, so let\u0027s see the 2 familiar functions."},{"Start":"00:19.725 ","End":"00:21.540","Text":"Let\u0027s take the first one,"},{"Start":"00:21.540 ","End":"00:23.205","Text":"just a quick sketch,"},{"Start":"00:23.205 ","End":"00:24.825","Text":"and let the function,"},{"Start":"00:24.825 ","End":"00:30.675","Text":"the first one be y equals x squared and the other one, y equals x."},{"Start":"00:30.675 ","End":"00:33.885","Text":"What I\u0027m going to do from these 2 functions,"},{"Start":"00:33.885 ","End":"00:36.765","Text":"I\u0027m going to make a third function."},{"Start":"00:36.765 ","End":"00:39.690","Text":"I\u0027ll throw it down here in the middle between"},{"Start":"00:39.690 ","End":"00:43.670","Text":"them, and this time, I\u0027m going to take a bit of this and a bit of"},{"Start":"00:43.670 ","End":"00:47.270","Text":"that, so I\u0027m going take this bit here of the y equals"},{"Start":"00:47.270 ","End":"00:52.235","Text":"x so this will be the bit from the y equals x."},{"Start":"00:52.235 ","End":"00:57.090","Text":"This bit here, I\u0027ll take from here, something like"},{"Start":"00:57.090 ","End":"01:04.020","Text":"this and I\u0027m going to emphasize by highlighting so this part of the parabola,"},{"Start":"01:04.020 ","End":"01:08.150","Text":"the y equals x squared I\u0027ll highlight in green,"},{"Start":"01:08.150 ","End":"01:11.840","Text":"and this is the same green bit."},{"Start":"01:11.840 ","End":"01:13.220","Text":"For the other bit,"},{"Start":"01:13.220 ","End":"01:17.875","Text":"I\u0027m taking this part of here and this will be this part here."},{"Start":"01:17.875 ","End":"01:22.390","Text":"The only question really is a question of the border between the"},{"Start":"01:22.390 ","End":"01:27.920","Text":"2 and the only thing I wanted to say is that the border which is at the origin,"},{"Start":"01:27.920 ","End":"01:31.535","Text":"0, 0, it actually sits on both of them"},{"Start":"01:31.535 ","End":"01:36.620","Text":"and it\u0027s not quite clear as to which side this farmer will deal with that later."},{"Start":"01:36.620 ","End":"01:41.150","Text":"We\u0027ll choose arbitrarily whether this point is on the yellow bit on the green bit."},{"Start":"01:41.150 ","End":"01:43.520","Text":"Now, how do I describe such a function?"},{"Start":"01:43.520 ","End":"01:46.040","Text":"It is a function, each x has a single value of y,"},{"Start":"01:46.040 ","End":"01:52.720","Text":"but it\u0027s made up of patches or I can see why they call it a hybrid function."},{"Start":"01:52.720 ","End":"01:54.910","Text":"Hybrid means a bit of this and a bit of that."},{"Start":"01:54.910 ","End":"01:57.985","Text":"The way we describe such a function is we use"},{"Start":"01:57.985 ","End":"02:02.665","Text":"a special notation using curly braces and I would say"},{"Start":"02:02.665 ","End":"02:06.780","Text":"that the function here would be y equals and"},{"Start":"02:06.780 ","End":"02:11.020","Text":"I use a curly brace here because there\u0027s 2 separate definitions."},{"Start":"02:11.020 ","End":"02:14.830","Text":"Sometimes, it\u0027s equal to y is equal to x,"},{"Start":"02:14.830 ","End":"02:18.550","Text":"and sometimes, y is equal to x squared."},{"Start":"02:18.550 ","End":"02:20.565","Text":"But which is which?"},{"Start":"02:20.565 ","End":"02:24.145","Text":"What we do is we write the word if,"},{"Start":"02:24.145 ","End":"02:28.705","Text":"and sometimes, people use a semicolon. I\u0027ll show you that in a minute."},{"Start":"02:28.705 ","End":"02:33.020","Text":"Now, when is it x, when we\u0027re here,"},{"Start":"02:33.020 ","End":"02:40.205","Text":"and this is when x is bigger than 0 is this part here and so I say if x is bigger than 0,"},{"Start":"02:40.205 ","End":"02:42.050","Text":"and when am I using the green bit?"},{"Start":"02:42.050 ","End":"02:51.690","Text":"When x is negative, when x is less than 0. So it\u0027s x squared if x is less than 0."},{"Start":"02:51.690 ","End":"02:55.490","Text":"As I said, we have a border dispute as to which side this goes on."},{"Start":"02:55.490 ","End":"02:58.400","Text":"Well, let\u0027s say it goes on the yellow side, so that"},{"Start":"02:58.400 ","End":"03:03.260","Text":"means that we say that if x is bigger or equal to 0,"},{"Start":"03:03.260 ","End":"03:07.325","Text":"so here, x equals 0."},{"Start":"03:07.325 ","End":"03:10.880","Text":"If x is bigger than 0 or equal to 0,"},{"Start":"03:10.880 ","End":"03:13.625","Text":"which means that this part is also a yellow."},{"Start":"03:13.625 ","End":"03:15.260","Text":"This green is not included,"},{"Start":"03:15.260 ","End":"03:17.750","Text":"this point is not on the green, it\u0027s on the yellow."},{"Start":"03:17.750 ","End":"03:20.840","Text":"That\u0027s a piecewise-defined function,"},{"Start":"03:20.840 ","End":"03:25.550","Text":"and it\u0027s very popular in math in colleges and universities,"},{"Start":"03:25.550 ","End":"03:29.555","Text":"but you hardly ever see it in high schools except perhaps, well,"},{"Start":"03:29.555 ","End":"03:31.460","Text":"from time to time, I\u0027ve seen it when they defined"},{"Start":"03:31.460 ","End":"03:34.925","Text":"absolute value of x, I\u0027ve seen this thing done in high school."},{"Start":"03:34.925 ","End":"03:38.900","Text":"Just as I say, there is an alternative notation."},{"Start":"03:38.900 ","End":"03:43.250","Text":"We can write that y equals x semicolon."},{"Start":"03:43.250 ","End":"03:44.975","Text":"I\u0027ve often seen that in the literature."},{"Start":"03:44.975 ","End":"03:50.075","Text":"Semicolon meaning if x on the occasions when x is bigger or equal to 0,"},{"Start":"03:50.075 ","End":"03:54.890","Text":"and x squared semicolon when x is less than 0,"},{"Start":"03:54.890 ","End":"03:57.995","Text":"just instead of the word if it makes no difference."},{"Start":"03:57.995 ","End":"04:00.410","Text":"I often say piecewise functions because you"},{"Start":"04:00.410 ","End":"04:02.480","Text":"get tired of saying piecewise-defined functions."},{"Start":"04:02.480 ","End":"04:04.730","Text":"That\u0027s it for the introduction."},{"Start":"04:04.730 ","End":"04:09.905","Text":"Let\u0027s give an example of how I substitute the values."},{"Start":"04:09.905 ","End":"04:11.960","Text":"But first, I wanted to actually use"},{"Start":"04:11.960 ","End":"04:14.280","Text":"functional notation instead of saying y equals I would"},{"Start":"04:14.280 ","End":"04:18.740","Text":"rather go to the f of x equals f of x equals similarly,"},{"Start":"04:18.740 ","End":"04:21.350","Text":"you could do that they\u0027re either y or f of x,"},{"Start":"04:21.350 ","End":"04:25.130","Text":"but I\u0027d like to use the functional notation because I want to ask you,"},{"Start":"04:25.130 ","End":"04:27.530","Text":"what is f of 4?"},{"Start":"04:27.530 ","End":"04:29.015","Text":"What is f of minus 4?"},{"Start":"04:29.015 ","End":"04:32.190","Text":"Let\u0027s go and do that now."},{"Start":"04:32.190 ","End":"04:38.070","Text":"I could ask, what is f of 4 equal to?"},{"Start":"04:38.070 ","End":"04:43.835","Text":"I take my value 4 and look for it and see which case fits."},{"Start":"04:43.835 ","End":"04:46.070","Text":"Is 4 fall under the less than 0?"},{"Start":"04:46.070 ","End":"04:48.500","Text":"No. Is 4 bigger or equal to 0?"},{"Start":"04:48.500 ","End":"04:52.700","Text":"Yes, so f of 4 is equal to 4."},{"Start":"04:52.700 ","End":"04:57.920","Text":"But for example, if I asked you what is f of minus 4?"},{"Start":"04:57.920 ","End":"05:00.500","Text":"Then I asked for minus 4,"},{"Start":"05:00.500 ","End":"05:02.240","Text":"which case does it fit?"},{"Start":"05:02.240 ","End":"05:06.010","Text":"It fits the less than 0 sign."},{"Start":"05:06.010 ","End":"05:14.680","Text":"This is equal to minus 4 squared, which is 16."},{"Start":"05:14.680 ","End":"05:17.915","Text":"Likewise, let\u0027s try another example."},{"Start":"05:17.915 ","End":"05:21.570","Text":"What is f of 2?"},{"Start":"05:21.740 ","End":"05:27.425","Text":"f of 2, and perhaps, I\u0027ll even throw this on here so I look, say 1,"},{"Start":"05:27.425 ","End":"05:33.050","Text":"2. 1, 2 is here and so I go up to the function and I"},{"Start":"05:33.050 ","End":"05:40.560","Text":"see 2 is on the x is bigger or equal to 0."},{"Start":"05:40.580 ","End":"05:44.025","Text":"That\u0027s where 2 falls so it\u0027s just equal to x,"},{"Start":"05:44.025 ","End":"05:51.720","Text":"which is 2 so f of 2 is 2 but f of minus"},{"Start":"05:51.720 ","End":"05:59.120","Text":"2 falls under the"},{"Start":"05:59.120 ","End":"06:02.510","Text":"less than 0 so I look at the formula x squared,"},{"Start":"06:02.510 ","End":"06:04.180","Text":"which is on the green."},{"Start":"06:04.180 ","End":"06:07.995","Text":"Here\u0027s minus 1 minus 2,"},{"Start":"06:07.995 ","End":"06:15.900","Text":"minus 2 squared, which is equal to 4."},{"Start":"06:15.900 ","End":"06:19.200","Text":"Finally, f of 0."},{"Start":"06:19.200 ","End":"06:20.940","Text":"We look it up less than 0?"},{"Start":"06:20.940 ","End":"06:22.935","Text":"No. Bigger or equal to 0?"},{"Start":"06:22.935 ","End":"06:25.215","Text":"Yes, so we use this formula,"},{"Start":"06:25.215 ","End":"06:28.130","Text":"plug-in 0, it\u0027s equal to 0, that\u0027s it."},{"Start":"06:28.130 ","End":"06:29.510","Text":"We\u0027re done for the theory part,"},{"Start":"06:29.510 ","End":"06:30.770","Text":"but please stay on."},{"Start":"06:30.770 ","End":"06:34.250","Text":"I\u0027m going to do a solved practice exercise."},{"Start":"06:34.250 ","End":"06:37.700","Text":"Now, we come to the exercise part of the clip."},{"Start":"06:37.700 ","End":"06:40.159","Text":"In this exercise, we\u0027re given the function."},{"Start":"06:40.159 ","End":"06:42.920","Text":"Which is obviously piecewise-defined."},{"Start":"06:42.920 ","End":"06:44.810","Text":"This is the subject of our lesson,"},{"Start":"06:44.810 ","End":"06:47.060","Text":"and it\u0027s defined as follows."},{"Start":"06:47.060 ","End":"06:49.855","Text":"For x between 0 and 4,"},{"Start":"06:49.855 ","End":"06:51.350","Text":"including 0 and 4,"},{"Start":"06:51.350 ","End":"06:57.560","Text":"we define f of x as x squared and for x which is less than 0 or negative,"},{"Start":"06:57.560 ","End":"07:00.650","Text":"we define f of x as minus x."},{"Start":"07:00.650 ","End":"07:07.070","Text":"Part a, we have to compute the value of f at 1,"},{"Start":"07:07.070 ","End":"07:10.535","Text":"4 minus 4, 0, and 7."},{"Start":"07:10.535 ","End":"07:13.760","Text":"Then we have to sketch the graph of f and finally,"},{"Start":"07:13.760 ","End":"07:18.970","Text":"we have to check if f is odd or even or neither, which means general."},{"Start":"07:18.970 ","End":"07:28.785","Text":"Part a first, let\u0027s see this should go quickly. So f of 1 first,"},{"Start":"07:28.785 ","End":"07:33.600","Text":"we look at 1 and we\u0027ll look at which range it\u0027s in and we see that 1"},{"Start":"07:33.600 ","End":"07:37.550","Text":"is between 0 and 4 so we go to the formula here,"},{"Start":"07:37.550 ","End":"07:43.125","Text":"1 squared and that\u0027s equal to 1."},{"Start":"07:43.125 ","End":"07:46.350","Text":"Next, f of 4."},{"Start":"07:46.350 ","End":"07:49.670","Text":"We check 4, it belongs in this range,"},{"Start":"07:49.670 ","End":"07:51.065","Text":"gets exactly at the edge,"},{"Start":"07:51.065 ","End":"07:56.090","Text":"but it still belongs to x less than or equal to 4."},{"Start":"07:56.090 ","End":"07:58.160","Text":"We go here, and again,"},{"Start":"07:58.160 ","End":"08:07.070","Text":"x squared is the formula, so f of 4 is 4 squared, which is 16."},{"Start":"08:07.070 ","End":"08:10.640","Text":"Next, f of 0, which ranged a 0 fall in we\u0027ll"},{"Start":"08:10.640 ","End":"08:14.360","Text":"obviously is not less than 0 and it\u0027s obviously, it\u0027s at the edge of this,"},{"Start":"08:14.360 ","End":"08:22.595","Text":"so it\u0027s again, x squared. So f of 0 is 0 squared, which is 0."},{"Start":"08:22.595 ","End":"08:25.860","Text":"Next, f of minus 4."},{"Start":"08:26.480 ","End":"08:30.610","Text":"f of minus 4, where does minus 4 belong?"},{"Start":"08:30.610 ","End":"08:34.250","Text":"It belongs in less than 0 clearly. So what we have,"},{"Start":"08:34.250 ","End":"08:41.300","Text":"it\u0027s minus x. So it\u0027s minus of minus 4, which is 4."},{"Start":"08:41.300 ","End":"08:45.680","Text":"Finally, what about f of 7?"},{"Start":"08:45.680 ","End":"08:55.795","Text":"Well, 7 is not less than 0 and it\u0027s not between 0 and 4."},{"Start":"08:55.795 ","End":"08:58.185","Text":"In fact, f of 7 is not defined."},{"Start":"08:58.185 ","End":"09:03.425","Text":"7 is not in lead domain of definition, so f of 7 is undefined."},{"Start":"09:03.425 ","End":"09:04.970","Text":"I don\u0027t know how to write that."},{"Start":"09:04.970 ","End":"09:07.790","Text":"You can write the word undefined or not in the domain."},{"Start":"09:07.790 ","End":"09:10.100","Text":"I\u0027ll just put an x here,"},{"Start":"09:10.100 ","End":"09:13.140","Text":"or maybe I\u0027ll write undefined."},{"Start":"09:16.710 ","End":"09:23.200","Text":"That finishes part a, and now, onto part b,"},{"Start":"09:23.200 ","End":"09:25.480","Text":"a little sketch of the graph of f."},{"Start":"09:25.480 ","End":"09:30.550","Text":"Now, what I\u0027d like to do in the sketch is just draw each piece separately."},{"Start":"09:30.550 ","End":"09:32.785","Text":"I\u0027ll draw some axes."},{"Start":"09:32.785 ","End":"09:38.160","Text":"The easier 1 is the graph of minus x."},{"Start":"09:38.160 ","End":"09:44.245","Text":"Now, the graph of minus x is a straight line and it goes through the origin,"},{"Start":"09:44.245 ","End":"09:46.780","Text":"sloping down at 45 degrees."},{"Start":"09:46.780 ","End":"09:50.710","Text":"But we only want to do this for x less than 0."},{"Start":"09:50.710 ","End":"09:55.495","Text":"First of all, see this is 0 and this is 4."},{"Start":"09:55.495 ","End":"09:59.290","Text":"So 0 less than or equal to x less than or equal to 4,"},{"Start":"09:59.290 ","End":"10:02.725","Text":"I shall color it with,"},{"Start":"10:02.725 ","End":"10:05.500","Text":"let\u0027s say, maybe red,"},{"Start":"10:05.500 ","End":"10:07.945","Text":"not bad, and make it stand out."},{"Start":"10:07.945 ","End":"10:10.540","Text":"This is x between 0 and 4,"},{"Start":"10:10.540 ","End":"10:16.525","Text":"and to just demonstrate that 4 is included here and here,"},{"Start":"10:16.525 ","End":"10:23.545","Text":"and in green for x less than 0."},{"Start":"10:23.545 ","End":"10:26.530","Text":"That will be all the way here."},{"Start":"10:26.530 ","End":"10:28.300","Text":"For this x,"},{"Start":"10:28.300 ","End":"10:31.585","Text":"it\u0027s going to be 1 way, and this continues on indefinitely."},{"Start":"10:31.585 ","End":"10:33.430","Text":"From 0-4, another way,"},{"Start":"10:33.430 ","End":"10:35.230","Text":"and from 4 onwards, it\u0027s not defined."},{"Start":"10:35.230 ","End":"10:40.870","Text":"This is outside. This is the area which I\u0027m going to leave uncolored."},{"Start":"10:40.870 ","End":"10:48.670","Text":"Minus x, you\u0027ve probably seen enough times to know how to plot a straight line."},{"Start":"10:48.670 ","End":"10:54.055","Text":"It\u0027s a straight line through the origin at 45 degrees."},{"Start":"10:54.055 ","End":"10:59.875","Text":"I just fixed that up a bit and labeled it as y equals minus x."},{"Start":"10:59.875 ","End":"11:06.265","Text":"The green area is the area where x is less than 0, and of course,"},{"Start":"11:06.265 ","End":"11:10.915","Text":"the red is the 0 less than or equal to x,"},{"Start":"11:10.915 ","End":"11:13.390","Text":"less than or equal to 4."},{"Start":"11:13.390 ","End":"11:15.340","Text":"In the red part,"},{"Start":"11:15.340 ","End":"11:21.080","Text":"the graph looks like the graph of y equals x squared."},{"Start":"11:22.440 ","End":"11:24.610","Text":"It\u0027s a bit of a parabola."},{"Start":"11:24.610 ","End":"11:26.590","Text":"I\u0027ll just label it."},{"Start":"11:26.590 ","End":"11:31.120","Text":"That here y equals x squared."},{"Start":"11:31.120 ","End":"11:33.820","Text":"It\u0027s defined in 2 different pieces."},{"Start":"11:33.820 ","End":"11:35.410","Text":"Here it looks like x squared,"},{"Start":"11:35.410 ","End":"11:38.090","Text":"here it looks like minus x."},{"Start":"11:41.640 ","End":"11:43.960","Text":"Here, it does not continue."},{"Start":"11:43.960 ","End":"11:46.390","Text":"This one goes on. I\u0027ll indicate that with an arrow."},{"Start":"11:46.390 ","End":"11:51.360","Text":"It goes on forever because the interval x less than 0 goes on forever,"},{"Start":"11:51.360 ","End":"11:53.760","Text":"but here, it stops at 4,"},{"Start":"11:53.760 ","End":"11:58.860","Text":"and this would actually be the point 4, 16 to be precise,"},{"Start":"11:58.860 ","End":"12:01.380","Text":"4, 16, and then it ends."},{"Start":"12:01.380 ","End":"12:04.425","Text":"They both meet at 0, 0."},{"Start":"12:04.425 ","End":"12:07.390","Text":"That\u0027s the rough sketch."},{"Start":"12:07.390 ","End":"12:12.430","Text":"In part c, we have to check if f is odd, even, or general."},{"Start":"12:12.430 ","End":"12:22.960","Text":"Let\u0027s just briefly recall what these 3 terms mean?"},{"Start":"12:22.960 ","End":"12:27.265","Text":"Even. A function f is even,"},{"Start":"12:27.265 ","End":"12:28.810","Text":"I\u0027m just writing it briefly,"},{"Start":"12:28.810 ","End":"12:34.720","Text":"if f of minus x is equal to"},{"Start":"12:34.720 ","End":"12:41.290","Text":"f of x for all x. I repeat, for all x."},{"Start":"12:41.290 ","End":"12:46.840","Text":"Which means that even if we find 1 single x for which this doesn\u0027t hold, it\u0027s not even."},{"Start":"12:46.840 ","End":"12:52.015","Text":"A similar thing happens with odd."},{"Start":"12:52.015 ","End":"12:56.140","Text":"In this case, we also try to compute f of minus x."},{"Start":"12:56.140 ","End":"13:01.660","Text":"But here, the requirement is that it equal minus f of x,"},{"Start":"13:01.660 ","End":"13:03.625","Text":"again, for all x."},{"Start":"13:03.625 ","End":"13:07.630","Text":"So if you find 1 single x for which this equality doesn\u0027t hold,"},{"Start":"13:07.630 ","End":"13:12.400","Text":"then it\u0027s not odd and a function in this context is called general."},{"Start":"13:12.400 ","End":"13:17.455","Text":"Basically, if it\u0027s neither odd nor even."},{"Start":"13:17.455 ","End":"13:19.555","Text":"What do we think?"},{"Start":"13:19.555 ","End":"13:21.115","Text":"I\u0027ll save you some time."},{"Start":"13:21.115 ","End":"13:24.400","Text":"I tried to prove it\u0027s even,"},{"Start":"13:24.400 ","End":"13:29.305","Text":"and if you start trying to with a split function like that,"},{"Start":"13:29.305 ","End":"13:31.600","Text":"it just doesn\u0027t work."},{"Start":"13:31.600 ","End":"13:36.910","Text":"I couldn\u0027t get it to work and I couldn\u0027t get it to work for being odd,"},{"Start":"13:36.910 ","End":"13:40.030","Text":"and in fact, I could see right away that there"},{"Start":"13:40.030 ","End":"13:42.775","Text":"are some examples that show that it\u0027s not odd or even."},{"Start":"13:42.775 ","End":"13:46.165","Text":"Why don\u0027t we just try some values of x?"},{"Start":"13:46.165 ","End":"13:47.575","Text":"What do you want to try?"},{"Start":"13:47.575 ","End":"13:49.270","Text":"Well, let\u0027s go for, I suggest,"},{"Start":"13:49.270 ","End":"13:52.300","Text":"somewhere between say 0 and 4."},{"Start":"13:52.300 ","End":"13:56.230","Text":"Let\u0027s try letting x equal 2."},{"Start":"13:56.230 ","End":"13:59.560","Text":"I\u0027ll just write that down."},{"Start":"13:59.560 ","End":"14:03.835","Text":"Let\u0027s just try x is equal to 2."},{"Start":"14:03.835 ","End":"14:06.040","Text":"There\u0027s 3 things we have to check."},{"Start":"14:06.040 ","End":"14:10.435","Text":"We have to check f of minus x,"},{"Start":"14:10.435 ","End":"14:15.580","Text":"we have to check f of x,"},{"Start":"14:15.580 ","End":"14:19.450","Text":"and we have to check minus f of x."},{"Start":"14:19.450 ","End":"14:22.120","Text":"This is what we have to check,"},{"Start":"14:22.120 ","End":"14:28.210","Text":"and we have to see that if this is not equal to either 1 of these,"},{"Start":"14:28.210 ","End":"14:34.645","Text":"then we\u0027re done by disproving the even and the odd and we\u0027re up to the general."},{"Start":"14:34.645 ","End":"14:37.630","Text":"In this case, x equals 2,"},{"Start":"14:37.630 ","End":"14:40.960","Text":"for f of x, we\u0027ll get,"},{"Start":"14:40.960 ","End":"14:45.055","Text":"I\u0027ll do it in a different color, say blue."},{"Start":"14:45.055 ","End":"14:51.805","Text":"First of all, let\u0027s do f of x. f of x is f of 2,"},{"Start":"14:51.805 ","End":"14:57.370","Text":"and 2 falls between 0 and 4 category,"},{"Start":"14:57.370 ","End":"14:59.200","Text":"so it\u0027s 2 squared."},{"Start":"14:59.200 ","End":"15:03.745","Text":"2 squared is 4."},{"Start":"15:03.745 ","End":"15:06.295","Text":"This bit is 4."},{"Start":"15:06.295 ","End":"15:12.920","Text":"Minus f of x is minus 4."},{"Start":"15:15.390 ","End":"15:19.225","Text":"F of minus x, we have first of all to look at minus x."},{"Start":"15:19.225 ","End":"15:23.455","Text":"Minus x would be minus 2,"},{"Start":"15:23.455 ","End":"15:27.235","Text":"and f of minus 2, then we look it up."},{"Start":"15:27.235 ","End":"15:30.295","Text":"Minus 2 falls under the less than 0 category,"},{"Start":"15:30.295 ","End":"15:32.980","Text":"so it\u0027s minus minus 2."},{"Start":"15:32.980 ","End":"15:37.720","Text":"If it\u0027s minus minus 2, then it\u0027s 2."},{"Start":"15:37.720 ","End":"15:41.530","Text":"Now, 2 is not equal to 4,"},{"Start":"15:41.530 ","End":"15:45.220","Text":"nor is 2 equal to negative 4."},{"Start":"15:45.220 ","End":"15:47.440","Text":"2 is not equal to either 1 of these,"},{"Start":"15:47.440 ","End":"15:51.580","Text":"so we\u0027re forced to come to the conclusion that we have"},{"Start":"15:51.580 ","End":"15:56.830","Text":"a general function by just letting x equals 2."},{"Start":"15:56.830 ","End":"16:02.120","Text":"I can\u0027t see why it wouldn\u0027t work."},{"Start":"16:03.750 ","End":"16:07.675","Text":"If we had tried another number like 1,"},{"Start":"16:07.675 ","End":"16:10.810","Text":"then f of 1,"},{"Start":"16:10.810 ","End":"16:14.170","Text":"these 2, it wouldn\u0027t."},{"Start":"16:14.170 ","End":"16:26.035","Text":"If we tried x equals 1,"},{"Start":"16:26.035 ","End":"16:32.350","Text":"then we would have had f of minus 1 is equal to"},{"Start":"16:32.350 ","End":"16:40.490","Text":"1 and f of 1 is also equal to 1."},{"Start":"16:42.570 ","End":"16:45.820","Text":"I\u0027ll say it again. If we had tried that,"},{"Start":"16:45.820 ","End":"16:48.410","Text":"say, x equals 1,"},{"Start":"16:48.840 ","End":"16:52.420","Text":"I\u0027m just writing this hypothetically."},{"Start":"16:52.420 ","End":"16:57.310","Text":"If x equals 1,"},{"Start":"16:57.310 ","End":"17:00.580","Text":"then the whole situation here would have become f of"},{"Start":"17:00.580 ","End":"17:07.810","Text":"1 is 1 squared is 1 here,"},{"Start":"17:07.810 ","End":"17:10.405","Text":"minus 1 would be here,"},{"Start":"17:10.405 ","End":"17:14.695","Text":"and f of minus 1 would be 1."},{"Start":"17:14.695 ","End":"17:16.720","Text":"You would get an equal."},{"Start":"17:16.720 ","End":"17:21.415","Text":"This would be a yes and this would be a no."},{"Start":"17:21.415 ","End":"17:26.110","Text":"But it doesn\u0027t mean that it\u0027s even,"},{"Start":"17:26.110 ","End":"17:29.110","Text":"it just means that we\u0027ve got lucky for 1 value."},{"Start":"17:29.110 ","End":"17:33.880","Text":"If you tried 1 and you were in doubt and you got 1 of these to be equal,"},{"Start":"17:33.880 ","End":"17:35.320","Text":"then you try another value."},{"Start":"17:35.320 ","End":"17:36.640","Text":"Then we\u0027d go on to try 2,"},{"Start":"17:36.640 ","End":"17:38.410","Text":"and we\u0027d find that in the case of 2,"},{"Start":"17:38.410 ","End":"17:42.040","Text":"all these 3 things are all different."},{"Start":"17:42.040 ","End":"17:46.660","Text":"That the 2 is not equal to the 4 and the 2 is not equal to the minus 4."},{"Start":"17:46.660 ","End":"17:52.405","Text":"It could be that sometimes you do get 1 of these 2 being equal but not consistently."},{"Start":"17:52.405 ","End":"17:55.270","Text":"Basically, that answers the third and last question."},{"Start":"17:55.270 ","End":"17:58.130","Text":"The function f is general, and we\u0027re done."}],"ID":1219},{"Watched":false,"Name":"Exercise 1","Duration":"2m 18s","ChapterTopicVideoID":5981,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this exercise, we have to sketch the graph of the function f of x equals 1,"},{"Start":"00:04.980 ","End":"00:07.815","Text":"plus absolute value of x minus 2."},{"Start":"00:07.815 ","End":"00:12.030","Text":"Let\u0027s write the definition of absolute value of x minus 2,"},{"Start":"00:12.030 ","End":"00:15.060","Text":"and this will give us a piece-wise definition of the graph."},{"Start":"00:15.060 ","End":"00:18.765","Text":"Now, remember, just reminding you that in general,"},{"Start":"00:18.765 ","End":"00:25.335","Text":"the absolute value of x is defined to be equal to x,"},{"Start":"00:25.335 ","End":"00:32.620","Text":"for x bigger or equal to 0 and minus x when x is less than 0."},{"Start":"00:32.620 ","End":"00:36.025","Text":"In our case, we have x minus 2,"},{"Start":"00:36.025 ","End":"00:42.600","Text":"the absolute value of x minus 2 is equal to x minus 2,"},{"Start":"00:42.600 ","End":"00:47.225","Text":"when x minus 2 is bigger or equal to 0,"},{"Start":"00:47.225 ","End":"00:54.020","Text":"and to minus x minus 2 when x minus 2 is less than 0."},{"Start":"00:54.020 ","End":"00:56.240","Text":"Let\u0027s simplify this a bit."},{"Start":"00:56.240 ","End":"01:05.305","Text":"We get the absolute value of x minus 2 is equal to it\u0027s x minus 2,"},{"Start":"01:05.305 ","End":"01:08.825","Text":"when x minus 2 is bigger or equal to 0."},{"Start":"01:08.825 ","End":"01:15.950","Text":"In other words, when x is bigger or equal to 2 and minus x minus 2,"},{"Start":"01:15.950 ","End":"01:18.530","Text":"which is 2 minus x."},{"Start":"01:18.530 ","End":"01:21.230","Text":"Whenever x minus 2 less than 0,"},{"Start":"01:21.230 ","End":"01:24.155","Text":"which means that x is less than 2."},{"Start":"01:24.155 ","End":"01:26.750","Text":"Now, that\u0027s just part of f of x."},{"Start":"01:26.750 ","End":"01:28.325","Text":"If we write it out,"},{"Start":"01:28.325 ","End":"01:35.225","Text":"we\u0027ll get that f of x is equal to piece-wise."},{"Start":"01:35.225 ","End":"01:38.780","Text":"We just take what\u0027s above and add the 1 plus."},{"Start":"01:38.780 ","End":"01:48.695","Text":"X minus 2 plus 1 is x minus 1 and that\u0027s true when x is bigger or equal to 2."},{"Start":"01:48.695 ","End":"01:51.095","Text":"2 minus x plus 1,"},{"Start":"01:51.095 ","End":"01:57.800","Text":"which is 3 minus x when x is less than 2."},{"Start":"01:57.800 ","End":"02:00.110","Text":"Okay? What we have is a function f,"},{"Start":"02:00.110 ","End":"02:03.410","Text":"which is defined piecewise in 2 bits and in each piece,"},{"Start":"02:03.410 ","End":"02:05.225","Text":"it\u0027s a linear graph."},{"Start":"02:05.225 ","End":"02:07.480","Text":"Let\u0027s start drawing."},{"Start":"02:07.480 ","End":"02:11.780","Text":"That\u0027s the y-axis, that\u0027s the x-axis."},{"Start":"02:11.780 ","End":"02:16.475","Text":"The most important point on the x-axis is the point x equals 2."},{"Start":"02:16.475 ","End":"02:20.330","Text":"That say that somewhere here and now"},{"Start":"02:20.330 ","End":"02:24.245","Text":"what we\u0027ll do is draw each of these lines separately."},{"Start":"02:24.245 ","End":"02:29.330","Text":"Let\u0027s say that we draw the first 1 in turquoise."},{"Start":"02:29.330 ","End":"02:31.130","Text":"F of x is x minus 1."},{"Start":"02:31.130 ","End":"02:32.420","Text":"There\u0027s many ways to do this."},{"Start":"02:32.420 ","End":"02:34.925","Text":"1 way is to substitute values."},{"Start":"02:34.925 ","End":"02:37.300","Text":"For example, when x is 2,"},{"Start":"02:37.300 ","End":"02:41.960","Text":"why is 1 and when x is 0,"},{"Start":"02:41.960 ","End":"02:44.220","Text":"y is minus 1."},{"Start":"02:44.220 ","End":"02:51.089","Text":"We get something like this and this is the point 2,"},{"Start":"02:51.089 ","End":"02:54.605","Text":"1, the other bit 3 minus x,"},{"Start":"02:54.605 ","End":"02:56.240","Text":"use a different color."},{"Start":"02:56.240 ","End":"02:57.830","Text":"Also we can substitute,"},{"Start":"02:57.830 ","End":"02:59.270","Text":"let\u0027s say when x is 0,"},{"Start":"02:59.270 ","End":"03:02.180","Text":"we\u0027ll get y is 3."},{"Start":"03:02.180 ","End":"03:07.639","Text":"And also we could take x equals 2 and then y would equal 1 also."},{"Start":"03:07.639 ","End":"03:12.785","Text":"Actually passes through the same point and we get something like this."},{"Start":"03:12.785 ","End":"03:18.454","Text":"This is the graph of 3 minus x,"},{"Start":"03:18.454 ","End":"03:27.340","Text":"say y equals 3 minus x and the other 1 was y equals x minus 1."},{"Start":"03:27.340 ","End":"03:31.940","Text":"Now what we have to do is to take into the account the domains."},{"Start":"03:31.940 ","End":"03:34.835","Text":"When x is bigger or equal to 2,"},{"Start":"03:34.835 ","End":"03:39.480","Text":"we\u0027re on the blue graph from here to here,"},{"Start":"03:39.480 ","End":"03:45.635","Text":"and then going on to infinity and the green bit when x is less than 2,"},{"Start":"03:45.635 ","End":"03:47.915","Text":"which does not include this point,"},{"Start":"03:47.915 ","End":"03:51.290","Text":"but this point is here anyway on the blue graph,"},{"Start":"03:51.290 ","End":"03:53.130","Text":"so we just continue,"},{"Start":"03:53.130 ","End":"04:01.170","Text":"and the answer to the question is this black part of the graph and we\u0027re done."}],"ID":5995},{"Watched":false,"Name":"Exercise 2","Duration":"2m 12s","ChapterTopicVideoID":5982,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to graph the function x"},{"Start":"00:03.270 ","End":"00:07.890","Text":"squared plus twice absolute value of x plus 1 plus 1."},{"Start":"00:07.890 ","End":"00:11.625","Text":"I\u0027d like to remind you what the absolute value means."},{"Start":"00:11.625 ","End":"00:15.990","Text":"In general, the absolute value of some number a is"},{"Start":"00:15.990 ","End":"00:21.539","Text":"equal to a itself if a is bigger or equal to 0,"},{"Start":"00:21.539 ","End":"00:26.250","Text":"and minus a if a is less than 0."},{"Start":"00:26.250 ","End":"00:29.910","Text":"In our case, the a is x plus 1."},{"Start":"00:29.910 ","End":"00:40.230","Text":"This is a piece-wise function and we can write it as f of x equals 2 pieces x"},{"Start":"00:40.230 ","End":"00:46.300","Text":"squared plus twice x plus 1 itself plus 1 in"},{"Start":"00:46.300 ","End":"00:53.615","Text":"the case that a bigger equal to 0 means x plus 1 bigger or equal to 0."},{"Start":"00:53.615 ","End":"00:57.935","Text":"The other piece is x squared, from the minus a,"},{"Start":"00:57.935 ","End":"01:02.980","Text":"we put minus twice x plus 1, plus 1."},{"Start":"01:02.980 ","End":"01:07.325","Text":"That\u0027s for the case where x plus 1 is less than 0."},{"Start":"01:07.325 ","End":"01:13.385","Text":"Let\u0027s simplify this a bit and write f of x equals."},{"Start":"01:13.385 ","End":"01:15.949","Text":"Just expanding this with simple algebra,"},{"Start":"01:15.949 ","End":"01:22.195","Text":"we get x squared plus 2x plus 3."},{"Start":"01:22.195 ","End":"01:24.700","Text":"In the case x plus 1 bigger or equal to 0,"},{"Start":"01:24.700 ","End":"01:29.080","Text":"which just means that x is bigger or equal to minus 1."},{"Start":"01:29.080 ","End":"01:34.530","Text":"Expanding this, we get x squared minus 2x minus 2 plus"},{"Start":"01:34.530 ","End":"01:40.390","Text":"1 minus 1 for the case where x is less than minus 1."},{"Start":"01:40.390 ","End":"01:43.810","Text":"Each of these 2 pieces is a parabola,"},{"Start":"01:43.810 ","End":"01:45.790","Text":"and we\u0027ll sketch each one separately."},{"Start":"01:45.790 ","End":"01:48.350","Text":"First, let\u0027s draw some axes."},{"Start":"01:48.350 ","End":"01:50.725","Text":"Let\u0027s look at the first parabola,"},{"Start":"01:50.725 ","End":"01:53.185","Text":"x squared plus 2x plus 3."},{"Start":"01:53.185 ","End":"01:57.235","Text":"To plot it, we need a couple of points to a 3."},{"Start":"01:57.235 ","End":"01:59.440","Text":"Usually, we take the apex,"},{"Start":"01:59.440 ","End":"02:02.610","Text":"which is where x is minus b over 2a."},{"Start":"02:02.610 ","End":"02:05.930","Text":"In this case, x equals minus 2 over 2,"},{"Start":"02:05.930 ","End":"02:07.490","Text":"which is minus 1."},{"Start":"02:07.490 ","End":"02:09.700","Text":"When x is minus 1,"},{"Start":"02:09.700 ","End":"02:11.655","Text":"that\u0027s one of the points we take,"},{"Start":"02:11.655 ","End":"02:17.300","Text":"we get that y is equal to minus 1 squared,"},{"Start":"02:17.300 ","End":"02:21.065","Text":"which is 1 minus 2 plus 3."},{"Start":"02:21.065 ","End":"02:22.820","Text":"That gives us 2."},{"Start":"02:22.820 ","End":"02:26.365","Text":"In other words, we want to plot the point minus 1, 2."},{"Start":"02:26.365 ","End":"02:30.520","Text":"Easiest thing to do to get another point would be to put x equals 0,"},{"Start":"02:30.520 ","End":"02:32.710","Text":"the intersection with the y axis,"},{"Start":"02:32.710 ","End":"02:36.230","Text":"and when x is 0, we get y equals 3."},{"Start":"02:36.230 ","End":"02:39.080","Text":"In other words, we plot the point 0,"},{"Start":"02:39.080 ","End":"02:43.025","Text":"3, and let\u0027s do that in 1 color."},{"Start":"02:43.025 ","End":"02:45.375","Text":"Minus 1, 2,"},{"Start":"02:45.375 ","End":"02:49.759","Text":"somewhere here, and 0, 3 here."},{"Start":"02:49.759 ","End":"02:52.640","Text":"This is the apex, so it\u0027s symmetrical about this."},{"Start":"02:52.640 ","End":"02:56.540","Text":"This is just a rough sketch, something like this."},{"Start":"02:56.540 ","End":"03:01.220","Text":"However, we don\u0027t want the complete parabola because the top part"},{"Start":"03:01.220 ","End":"03:05.975","Text":"only applies when x is bigger or equal to minus 1,"},{"Start":"03:05.975 ","End":"03:08.375","Text":"which is from minus 1 onward."},{"Start":"03:08.375 ","End":"03:10.940","Text":"In other words, what we need to do,"},{"Start":"03:10.940 ","End":"03:13.834","Text":"is to erase the parts we don\u0027t want."},{"Start":"03:13.834 ","End":"03:16.850","Text":"There we go. Tidied up the graph a bit."},{"Start":"03:16.850 ","End":"03:18.380","Text":"It wouldn\u0027t look so good."},{"Start":"03:18.380 ","End":"03:20.555","Text":"Let\u0027s continue to the second half,"},{"Start":"03:20.555 ","End":"03:23.015","Text":"the x squared minus 2x minus 1,"},{"Start":"03:23.015 ","End":"03:24.979","Text":"and plot a few points."},{"Start":"03:24.979 ","End":"03:26.900","Text":"I always like to take the apex,"},{"Start":"03:26.900 ","End":"03:28.670","Text":"which is minus b over 2a,"},{"Start":"03:28.670 ","End":"03:32.125","Text":"in this case, x equals 1."},{"Start":"03:32.125 ","End":"03:34.700","Text":"That gives us, if we compute in our heads,"},{"Start":"03:34.700 ","End":"03:38.390","Text":"y is 1 squared minus 2 minus 1,"},{"Start":"03:38.390 ","End":"03:41.035","Text":"and that gives us minus 2."},{"Start":"03:41.035 ","End":"03:42.270","Text":"In other words,"},{"Start":"03:42.270 ","End":"03:45.545","Text":"we want to plot the point 1 minus 2."},{"Start":"03:45.545 ","End":"03:50.730","Text":"Another easy point to plot is the interception with the y-axis"},{"Start":"03:50.730 ","End":"03:56.060","Text":"where x equals 0 and we immediately see that that gives us y equals minus 1,"},{"Start":"03:56.060 ","End":"03:59.705","Text":"so we have another, 0, minus 1."},{"Start":"03:59.705 ","End":"04:01.954","Text":"But in the case of a piece-wise function,"},{"Start":"04:01.954 ","End":"04:07.860","Text":"I also like to take the value of the border between 2 domains."},{"Start":"04:09.340 ","End":"04:12.590","Text":"In the previous case, I already had minus 1,"},{"Start":"04:12.590 ","End":"04:14.045","Text":"but let\u0027s take it here."},{"Start":"04:14.045 ","End":"04:16.180","Text":"X equals minus 1,"},{"Start":"04:16.180 ","End":"04:19.880","Text":"and substitute, and I\u0027ll just give you the answer."},{"Start":"04:19.880 ","End":"04:21.050","Text":"Y comes out too,"},{"Start":"04:21.050 ","End":"04:22.580","Text":"you can do the computation."},{"Start":"04:22.580 ","End":"04:25.630","Text":"We get minus 1, 2."},{"Start":"04:25.630 ","End":"04:29.480","Text":"Let\u0027s plot these points and the graph in a different color."},{"Start":"04:29.480 ","End":"04:31.545","Text":"Let\u0027s choose blue this time."},{"Start":"04:31.545 ","End":"04:34.850","Text":"I drew it for you and didn\u0027t come out the greatest."},{"Start":"04:34.850 ","End":"04:39.245","Text":"That\u0027s just a sketch. Here\u0027s the point 1 minus 2."},{"Start":"04:39.245 ","End":"04:41.850","Text":"Here\u0027s the point 0 minus 1."},{"Start":"04:41.850 ","End":"04:43.365","Text":"The minus 1,"},{"Start":"04:43.365 ","End":"04:44.900","Text":"2 comes out here,"},{"Start":"04:44.900 ","End":"04:48.230","Text":"which is exactly the same as the green point we had there."},{"Start":"04:48.230 ","End":"04:51.505","Text":"That was the apex. It\u0027s symmetrical here."},{"Start":"04:51.505 ","End":"04:54.855","Text":"Just draw a line through this."},{"Start":"04:54.855 ","End":"05:01.330","Text":"Again, we have to remember that this is defined for x less than minus 1."},{"Start":"05:01.330 ","End":"05:05.285","Text":"We have to only take the part from minus 1 and to the left,"},{"Start":"05:05.285 ","End":"05:07.850","Text":"which means we need to erase all the bits"},{"Start":"05:07.850 ","End":"05:11.225","Text":"from minus 1 to the right, and I just did that."},{"Start":"05:11.225 ","End":"05:13.400","Text":"This is a rough sketch of the answer."},{"Start":"05:13.400 ","End":"05:14.450","Text":"Could have been drawn better,"},{"Start":"05:14.450 ","End":"05:18.160","Text":"but that\u0027s the general idea, and we\u0027re done."}],"ID":5996},{"Watched":false,"Name":"Exercise 3","Duration":"2m 16s","ChapterTopicVideoID":5983,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to graph the function f of"},{"Start":"00:03.060 ","End":"00:07.590","Text":"x equals the absolute value of x squared plus x minus 2."},{"Start":"00:07.590 ","End":"00:10.545","Text":"I want to remind you what absolute value means."},{"Start":"00:10.545 ","End":"00:14.310","Text":"It\u0027s defined piece wise that the absolute value of some number,"},{"Start":"00:14.310 ","End":"00:21.135","Text":"let\u0027s say a is equal to a whenever a is bigger or equal to 0,"},{"Start":"00:21.135 ","End":"00:26.070","Text":"and minus a whenever a is less than 0."},{"Start":"00:26.070 ","End":"00:30.165","Text":"In our case, if we write the function in piece wise form,"},{"Start":"00:30.165 ","End":"00:32.820","Text":"we get f of x is equal to,"},{"Start":"00:32.820 ","End":"00:35.910","Text":"here a is this whole x squared plus x minus 2."},{"Start":"00:35.910 ","End":"00:40.905","Text":"It\u0027s equal to the x squared plus x minus 2 itself."},{"Start":"00:40.905 ","End":"00:47.540","Text":"When x squared plus x minus 2 is bigger or equal to 0,"},{"Start":"00:47.540 ","End":"00:50.000","Text":"but minus the same thing."},{"Start":"00:50.000 ","End":"00:55.820","Text":"Minus x squared plus x minus 2 whenever this is less than 0."},{"Start":"00:55.820 ","End":"01:02.695","Text":"In other words, x squared plus x minus 2 is less than 0."},{"Start":"01:02.695 ","End":"01:06.445","Text":"What we have to do is solve 2 inequalities,"},{"Start":"01:06.445 ","End":"01:07.850","Text":"x squared plus x minus 2,"},{"Start":"01:07.850 ","End":"01:11.035","Text":"bigger or equal to 0 or less than 0."},{"Start":"01:11.035 ","End":"01:15.050","Text":"The way we do it is by solving the quadratic equation,"},{"Start":"01:15.050 ","End":"01:23.625","Text":"x squared plus x minus 2 equals 0 and using the roots to help us solve the inequality."},{"Start":"01:23.625 ","End":"01:25.750","Text":"Now, I\u0027m assuming you know how to solve"},{"Start":"01:25.750 ","End":"01:28.345","Text":"quadratic equations and I won\u0027t do all the work here."},{"Start":"01:28.345 ","End":"01:33.910","Text":"I\u0027ll come straight to the answer that x is equal to either 1 or minus 2."},{"Start":"01:33.910 ","End":"01:35.805","Text":"Okay, let\u0027s start sketching."},{"Start":"01:35.805 ","End":"01:38.290","Text":"It\u0027s piece wise, so we\u0027ll do each piece separately."},{"Start":"01:38.290 ","End":"01:41.065","Text":"The top 1 is x squared plus x minus 2."},{"Start":"01:41.065 ","End":"01:42.730","Text":"It\u0027s a quadratic function,"},{"Start":"01:42.730 ","End":"01:46.720","Text":"so it\u0027s a parabola and the coefficient of x squared is positive,"},{"Start":"01:46.720 ","End":"01:49.405","Text":"so we have an upward facing parabola."},{"Start":"01:49.405 ","End":"01:53.635","Text":"We also know the roots are 1 and minus 2."},{"Start":"01:53.635 ","End":"01:56.200","Text":"We can do a rough sketch,"},{"Start":"01:56.200 ","End":"01:59.525","Text":"1 minus 2,"},{"Start":"01:59.525 ","End":"02:04.075","Text":"continue here and continue on from here."},{"Start":"02:04.075 ","End":"02:05.790","Text":"Something like this."},{"Start":"02:05.790 ","End":"02:07.770","Text":"That\u0027s the complete parabola,"},{"Start":"02:07.770 ","End":"02:10.055","Text":"but we have to confine ourselves to"},{"Start":"02:10.055 ","End":"02:15.605","Text":"where it\u0027s defined x less than or equal to minus 2 or x bigger or equal to 1."},{"Start":"02:15.605 ","End":"02:20.539","Text":"Other words from minus 2 and in the downward direction,"},{"Start":"02:20.539 ","End":"02:22.055","Text":"or from 1,"},{"Start":"02:22.055 ","End":"02:25.254","Text":"then the rightward direction."},{"Start":"02:25.254 ","End":"02:28.745","Text":"It means we have to just erase the middle bit."},{"Start":"02:28.745 ","End":"02:31.415","Text":"Now for the bottom part,"},{"Start":"02:31.415 ","End":"02:33.785","Text":"this time it\u0027s also a parabola,"},{"Start":"02:33.785 ","End":"02:35.630","Text":"but it\u0027s downward facing because"},{"Start":"02:35.630 ","End":"02:38.869","Text":"the coefficient of x squared is minus 1, which is negative."},{"Start":"02:38.869 ","End":"02:41.300","Text":"It\u0027s a downward facing parabola,"},{"Start":"02:41.300 ","End":"02:43.390","Text":"and it also has the same roots."},{"Start":"02:43.390 ","End":"02:44.990","Text":"Because if we make this equal to 0,"},{"Start":"02:44.990 ","End":"02:47.360","Text":"it\u0027s the same as making its negative equal to 0."},{"Start":"02:47.360 ","End":"02:51.290","Text":"We need a downward facing parabola that goes through 1 and minus 2."},{"Start":"02:51.290 ","End":"02:53.780","Text":"That\u0027s the same points as we had before."},{"Start":"02:53.780 ","End":"02:57.475","Text":"The complete parabola would look something,"},{"Start":"02:57.475 ","End":"03:01.250","Text":"and this time, we also have to look at the domain."},{"Start":"03:01.250 ","End":"03:06.830","Text":"It\u0027s only defined this way between minus 2 and 1, not including."},{"Start":"03:06.830 ","End":"03:08.975","Text":"In other words, the bit from,"},{"Start":"03:08.975 ","End":"03:11.300","Text":"from here to here,"},{"Start":"03:11.300 ","End":"03:14.015","Text":"along here, but not the rest of it."},{"Start":"03:14.015 ","End":"03:17.650","Text":"We need to erase this part and this part,"},{"Start":"03:17.650 ","End":"03:20.780","Text":"which gives us this graph, could be prettier,"},{"Start":"03:20.780 ","End":"03:25.080","Text":"but you get the general idea. We\u0027re done."}],"ID":5997},{"Watched":false,"Name":"Exercise 4 part a","Duration":"4m 38s","ChapterTopicVideoID":6140,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this exercise, we\u0027re given a couple of graphs."},{"Start":"00:02.670 ","End":"00:04.995","Text":"For each of them, we have to find the formula,"},{"Start":"00:04.995 ","End":"00:07.890","Text":"which means an equation y in terms of x."},{"Start":"00:07.890 ","End":"00:10.320","Text":"We\u0027ll start with the one on the left."},{"Start":"00:10.320 ","End":"00:15.480","Text":"It looks like a piece-wise graph because there\u0027s 2 separate bits of line."},{"Start":"00:15.480 ","End":"00:16.890","Text":"I\u0027ll highlight them."},{"Start":"00:16.890 ","End":"00:18.910","Text":"This is one bit,"},{"Start":"00:18.910 ","End":"00:20.900","Text":"put it in turquoise,"},{"Start":"00:20.900 ","End":"00:25.430","Text":"and the other bit indicates in green."},{"Start":"00:25.430 ","End":"00:27.670","Text":"Those are the 2 separate bits."},{"Start":"00:27.670 ","End":"00:30.060","Text":"We\u0027re going to have a piecewise definition."},{"Start":"00:30.060 ","End":"00:32.070","Text":"Not entirely clear from the picture,"},{"Start":"00:32.070 ","End":"00:39.030","Text":"but let\u0027s assume that this point here and this point here are on the graph."},{"Start":"00:39.030 ","End":"00:41.580","Text":"We also have a special point here."},{"Start":"00:41.580 ","End":"00:45.540","Text":"We\u0027re going to find out the formula for each of the pieces,"},{"Start":"00:45.540 ","End":"00:47.685","Text":"the blue and the green separately."},{"Start":"00:47.685 ","End":"00:50.880","Text":"The domains for the blue piece,"},{"Start":"00:50.880 ","End":"00:58.560","Text":"we\u0027re going to see that 0 less than or equal to x and let\u0027s say we include the 1."},{"Start":"00:58.560 ","End":"01:00.870","Text":"For the other bit, well,"},{"Start":"01:00.870 ","End":"01:02.010","Text":"we\u0027ve used the 1 here,"},{"Start":"01:02.010 ","End":"01:05.715","Text":"so we have to say 1 less than x,"},{"Start":"01:05.715 ","End":"01:07.680","Text":"less than or equal to 2."},{"Start":"01:07.680 ","End":"01:11.265","Text":"Now, we just have to find the formula for each piece."},{"Start":"01:11.265 ","End":"01:17.640","Text":"Line is y equals ax plus b,"},{"Start":"01:17.640 ","End":"01:20.055","Text":"and we have to find out a and b,"},{"Start":"01:20.055 ","End":"01:23.460","Text":"first for the blue line and then for the green line."},{"Start":"01:23.460 ","End":"01:25.740","Text":"We\u0027ll start with the blue one."},{"Start":"01:25.740 ","End":"01:30.780","Text":"Notice that the blue one passes through the points 0,"},{"Start":"01:30.780 ","End":"01:34.845","Text":"0 and 1, 1."},{"Start":"01:34.845 ","End":"01:37.650","Text":"What we can do is substitute these x,"},{"Start":"01:37.650 ","End":"01:41.775","Text":"y pairs in this and get a couple of equations in a and b."},{"Start":"01:41.775 ","End":"01:43.320","Text":"Putting x equals 0,"},{"Start":"01:43.320 ","End":"01:44.910","Text":"y equals 0,"},{"Start":"01:44.910 ","End":"01:54.405","Text":"we get that 0 equals a times 0 plus b, 1 equation."},{"Start":"01:54.405 ","End":"01:55.770","Text":"From this point,"},{"Start":"01:55.770 ","End":"02:03.885","Text":"we get that y is 1 equals a times 1 plus b."},{"Start":"02:03.885 ","End":"02:05.505","Text":"The solution is clear."},{"Start":"02:05.505 ","End":"02:06.720","Text":"From the first one,"},{"Start":"02:06.720 ","End":"02:08.355","Text":"you get that b is 0,"},{"Start":"02:08.355 ","End":"02:10.050","Text":"and if you put b equals 0 here,"},{"Start":"02:10.050 ","End":"02:11.655","Text":"you get a equals 1."},{"Start":"02:11.655 ","End":"02:18.825","Text":"We get that a equals 1, b equals 0."},{"Start":"02:18.825 ","End":"02:23.445","Text":"We put this into the y equals ax plus b."},{"Start":"02:23.445 ","End":"02:26.340","Text":"We get that the b is 0 and a is 1."},{"Start":"02:26.340 ","End":"02:32.085","Text":"We just get that y is equal to x."},{"Start":"02:32.085 ","End":"02:34.470","Text":"That\u0027s the blue line,"},{"Start":"02:34.470 ","End":"02:36.525","Text":"I\u0027ll just highlight it,"},{"Start":"02:36.525 ","End":"02:39.045","Text":"and this is the domain."},{"Start":"02:39.045 ","End":"02:41.145","Text":"That\u0027s one part."},{"Start":"02:41.145 ","End":"02:44.295","Text":"Now, let\u0027s go onto the green bit."},{"Start":"02:44.295 ","End":"02:46.905","Text":"Again, y is ax plus b."},{"Start":"02:46.905 ","End":"02:49.500","Text":"But this time, we have the points 1,"},{"Start":"02:49.500 ","End":"02:54.075","Text":"1, and the point 2, 0."},{"Start":"02:54.075 ","End":"02:56.070","Text":"We take this time the points 1,"},{"Start":"02:56.070 ","End":"02:57.510","Text":"1 and 2, 0,"},{"Start":"02:57.510 ","End":"03:01.590","Text":"substitute them in y equals ax plus b."},{"Start":"03:01.590 ","End":"03:03.960","Text":"This time, we get 1,"},{"Start":"03:03.960 ","End":"03:06.855","Text":"1 gives us the same equation as here,"},{"Start":"03:06.855 ","End":"03:12.825","Text":"so 1 is equal to a times 1 plus b."},{"Start":"03:12.825 ","End":"03:17.370","Text":"The 2, 0 gives us that 0,"},{"Start":"03:17.370 ","End":"03:18.600","Text":"which is the y,"},{"Start":"03:18.600 ","End":"03:23.235","Text":"is a times 2 plus b,"},{"Start":"03:23.235 ","End":"03:26.235","Text":"2 equations and 2 unknowns."},{"Start":"03:26.235 ","End":"03:30.225","Text":"If we subtract the first equation from the second,"},{"Start":"03:30.225 ","End":"03:35.250","Text":"we see that a is minus 1."},{"Start":"03:35.250 ","End":"03:37.860","Text":"If you put a equals minus 1 here,"},{"Start":"03:37.860 ","End":"03:40.110","Text":"I get 1 equals minus 1 plus b."},{"Start":"03:40.110 ","End":"03:43.095","Text":"In other words, b equals 2."},{"Start":"03:43.095 ","End":"03:44.760","Text":"Putting that into the equation,"},{"Start":"03:44.760 ","End":"03:46.800","Text":"y equals ax plus b,"},{"Start":"03:46.800 ","End":"03:53.595","Text":"we get that y equals minus x plus 2,"},{"Start":"03:53.595 ","End":"03:58.080","Text":"and that\u0027s the green bit with domain from here."},{"Start":"03:58.080 ","End":"04:04.590","Text":"All that remains is just to write it up that y equals x for x is between 0 and 1,"},{"Start":"04:04.590 ","End":"04:08.505","Text":"and that y is given from this formula for x from here,"},{"Start":"04:08.505 ","End":"04:10.365","Text":"I\u0027ll just write it up."},{"Start":"04:10.365 ","End":"04:14.280","Text":"We get that y equals,"},{"Start":"04:14.280 ","End":"04:16.905","Text":"the blue bit is x,"},{"Start":"04:16.905 ","End":"04:21.210","Text":"and this is true from 0 less than or equal to x,"},{"Start":"04:21.210 ","End":"04:23.445","Text":"less than or equal to 1."},{"Start":"04:23.445 ","End":"04:28.185","Text":"The green bit minus x plus 2,"},{"Start":"04:28.185 ","End":"04:32.910","Text":"and this holds for one less than x,"},{"Start":"04:32.910 ","End":"04:35.580","Text":"less than or equal 2."},{"Start":"04:35.580 ","End":"04:39.130","Text":"We\u0027re done for the first graph."}],"ID":6152},{"Watched":false,"Name":"Exercise 4 part b","Duration":"3m 46s","ChapterTopicVideoID":6139,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Next we\u0027ll work on the second graph,"},{"Start":"00:02.730 ","End":"00:03.930","Text":"the 1 on the right."},{"Start":"00:03.930 ","End":"00:07.140","Text":"Like before, let\u0027s mark some of the interesting points,"},{"Start":"00:07.140 ","End":"00:09.915","Text":"this 1 is 0, 0,"},{"Start":"00:09.915 ","End":"00:11.969","Text":"this 1, although it\u0027s not included,"},{"Start":"00:11.969 ","End":"00:14.190","Text":"is the point 1,2,"},{"Start":"00:14.190 ","End":"00:16.125","Text":"x is 1, y is 2."},{"Start":"00:16.125 ","End":"00:20.250","Text":"This is the point where x is 1 and y is 0."},{"Start":"00:20.250 ","End":"00:22.470","Text":"This point, although it\u0027s a hollow circle,"},{"Start":"00:22.470 ","End":"00:25.890","Text":"not included nevertheless we do know its coordinates,"},{"Start":"00:25.890 ","End":"00:29.460","Text":"x is 2 and y is minus 1."},{"Start":"00:29.460 ","End":"00:32.265","Text":"As before, we have 2 separate pieces."},{"Start":"00:32.265 ","End":"00:34.050","Text":"It looks like a piecewise definition,"},{"Start":"00:34.050 ","End":"00:35.760","Text":"so I\u0027ll use the same colors."},{"Start":"00:35.760 ","End":"00:38.915","Text":"The first piece, we\u0027ll call the blue piece,"},{"Start":"00:38.915 ","End":"00:42.679","Text":"and the second piece will be the green."},{"Start":"00:42.679 ","End":"00:44.615","Text":"We can write the domains."},{"Start":"00:44.615 ","End":"00:47.500","Text":"The green 1 goes from 1-2,"},{"Start":"00:47.500 ","End":"00:49.905","Text":"including the 1 and not including the 2,"},{"Start":"00:49.905 ","End":"00:55.530","Text":"which means 1 less than equal to x less than 2."},{"Start":"00:55.530 ","End":"00:58.680","Text":"The blue domain is from 0-1,"},{"Start":"00:58.680 ","End":"01:01.305","Text":"including the 0, but not including the 1."},{"Start":"01:01.305 ","End":"01:05.695","Text":"0 less than or equal to x less than 1."},{"Start":"01:05.695 ","End":"01:09.935","Text":"What we need to do now is find the formula for each of the 2 pieces."},{"Start":"01:09.935 ","End":"01:12.470","Text":"Using the same technique as before,"},{"Start":"01:12.470 ","End":"01:17.060","Text":"remembering that the general form of a line is y equals ax plus b."},{"Start":"01:17.060 ","End":"01:18.200","Text":"For the blue piece,"},{"Start":"01:18.200 ","End":"01:20.305","Text":"we substitute these 2 points,"},{"Start":"01:20.305 ","End":"01:22.850","Text":"the 00 and the 12."},{"Start":"01:22.850 ","End":"01:25.205","Text":"We get from 00,"},{"Start":"01:25.205 ","End":"01:31.760","Text":"we get that 0 equals a times 0 plus b."},{"Start":"01:31.760 ","End":"01:38.225","Text":"The 1, 2 gives us the 2 equals a times 1 plus b."},{"Start":"01:38.225 ","End":"01:41.660","Text":"First of these 2 equations just simply gives us that b is"},{"Start":"01:41.660 ","End":"01:45.860","Text":"0 and putting b is 0 here gives us as a is 2."},{"Start":"01:45.860 ","End":"01:48.460","Text":"In other words, a equals 2,"},{"Start":"01:48.460 ","End":"01:55.850","Text":"b equals 0 and the line becomes y equals ax plus b,"},{"Start":"01:55.850 ","End":"01:59.000","Text":"which is 2x plus nothing."},{"Start":"01:59.000 ","End":"02:01.565","Text":"That\u0027s this, blue line here."},{"Start":"02:01.565 ","End":"02:03.710","Text":"Let me highlight it."},{"Start":"02:03.710 ","End":"02:05.470","Text":"Y equals 2x,"},{"Start":"02:05.470 ","End":"02:07.865","Text":"That\u0027s the blue, and it\u0027s defined here."},{"Start":"02:07.865 ","End":"02:11.210","Text":"Next, we\u0027ll go to the green line."},{"Start":"02:11.210 ","End":"02:18.155","Text":"Here we have to substitute 1,0 and 2 minus 1 into the same y equals ax plus b."},{"Start":"02:18.155 ","End":"02:25.355","Text":"The first 1 will give us that 0 equals a times 1 plus b."},{"Start":"02:25.355 ","End":"02:33.735","Text":"The other point will give us the minus 1 is equal to a times 2 plus b."},{"Start":"02:33.735 ","End":"02:36.589","Text":"Subtracting the first equation from the second,"},{"Start":"02:36.589 ","End":"02:41.360","Text":"I see that a is equal to minus 1."},{"Start":"02:41.360 ","End":"02:43.910","Text":"If I substitute a is minus 1,"},{"Start":"02:43.910 ","End":"02:46.220","Text":"I\u0027ll see that b equals 1."},{"Start":"02:46.220 ","End":"02:49.015","Text":"Writing that into the y equals ax plus b,"},{"Start":"02:49.015 ","End":"02:54.630","Text":"we get y equals minus x plus 1."},{"Start":"02:54.630 ","End":"02:57.195","Text":"That\u0027s the green bit."},{"Start":"02:57.195 ","End":"02:59.720","Text":"Now we have to just put everything together."},{"Start":"02:59.720 ","End":"03:06.760","Text":"The y equals minus x plus 1 came with 1 less than or equal to x less than 2."},{"Start":"03:06.760 ","End":"03:14.120","Text":"The y equals 2x came from the section where x is bigger or equal to 0 and less than 1."},{"Start":"03:14.120 ","End":"03:16.535","Text":"Let\u0027s just write all that up."},{"Start":"03:16.535 ","End":"03:21.500","Text":"What we get is that y equals piecewise,"},{"Start":"03:21.500 ","End":"03:24.430","Text":"so it\u0027s equal to 2x."},{"Start":"03:24.430 ","End":"03:28.410","Text":"That was for 0 less than or equal to x less than 1."},{"Start":"03:28.410 ","End":"03:32.720","Text":"0, less than or equal to x less than 1."},{"Start":"03:32.720 ","End":"03:38.445","Text":"It was equal to the green bit minus x plus 1."},{"Start":"03:38.445 ","End":"03:42.590","Text":"This was for 1 less than or equal to x less than 2."},{"Start":"03:42.590 ","End":"03:44.585","Text":"That\u0027s the answer for the second part."},{"Start":"03:44.585 ","End":"03:47.280","Text":"We\u0027re done with this exercise."}],"ID":6151},{"Watched":false,"Name":"Exercise 5","Duration":"12m 58s","ChapterTopicVideoID":31661,"CourseChapterTopicPlaylistID":1186,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.140 ","End":"00:06.750","Text":"In this video, we\u0027re going to beexpressing the function F of X, which"},{"Start":"00:06.750 ","End":"00:11.760","Text":"is equal to the absolute value of xplus the absolute value of X minus"},{"Start":"00:11.760 ","End":"00:18.300","Text":"two, plus the absolute value of X minusthree as a piece-wise defined function"},{"Start":"00:18.810 ","End":"00:21.420","Text":"without these absolute value bars."},{"Start":"00:21.810 ","End":"00:25.830","Text":"So what we mean by that whenwe say peace wise is Essent."},{"Start":"00:26.370 ","End":"00:31.050","Text":"Working out how this plotlooks within specific domains."},{"Start":"00:31.500 ","End":"00:36.300","Text":"So the trick to answering these sortsof questions, or I say trick the"},{"Start":"00:36.300 ","End":"00:42.840","Text":"method, is really to just see howthe various parts of this function"},{"Start":"00:43.560 ","End":"00:45.390","Text":"work within specific domains."},{"Start":"00:46.050 ","End":"00:51.769","Text":"So if we call this partcomponent one and oops."},{"Start":"00:53.820 ","End":"00:58.410","Text":"and this part, component two,and then this component three."},{"Start":"00:59.130 ","End":"01:04.120","Text":"Then ultimately we just need to knowhow these parts act within a specific"},{"Start":"01:04.125 ","End":"01:08.550","Text":"domain, and then we can combine themall together and that\u0027s what\u0027s going"},{"Start":"01:08.555 ","End":"01:13.040","Text":"to give us the piece-wise definedfunction without absolute values."},{"Start":"01:14.025 ","End":"01:20.445","Text":"Once you\u0027ve seen this method, once, youcan ultimately apply it to any situation"},{"Start":"01:20.445 ","End":"01:25.035","Text":"like this again, even if the function is abit more complex and you know, you say you"},{"Start":"01:25.035 ","End":"01:37.245","Text":"had something like F of X is equal to, Idon\u0027t know, some mod of CX plus two mod X"},{"Start":"01:37.245 ","End":"01:40.985","Text":"minus three, or you know, plus da, da, da."},{"Start":"01:40.990 ","End":"01:42.825","Text":"The method will remain exactly the."},{"Start":"01:43.830 ","End":"01:46.380","Text":". So how are we gonna tackle this?"},{"Start":"01:46.380 ","End":"01:52.320","Text":"Well, let\u0027s just look at this firstcomponent, which is just modus of X."},{"Start":"01:52.350 ","End":"01:54.480","Text":"So modus of X."},{"Start":"01:54.780 ","End":"01:57.030","Text":"Well, how is this defined?"},{"Start":"01:57.060 ","End":"02:02.760","Text":"Well, you may have seen this one before,and that\u0027s just modus of X is equal to X"},{"Start":"02:02.760 ","End":"02:09.220","Text":"when X is greater than or equal to zero,and it\u0027s equal to negative x when x."},{"Start":"02:09.780 ","End":"02:10.859","Text":"Less than zero."},{"Start":"02:11.399 ","End":"02:13.320","Text":"And what\u0027s the justification for that?"},{"Start":"02:13.380 ","End":"02:18.690","Text":"Well, we need to make sure thatwhatever value X is, when we"},{"Start":"02:18.690 ","End":"02:22.980","Text":"take the absolute value of it, itreturns back a positive result."},{"Start":"02:23.640 ","End":"02:27.390","Text":"So if X is already, orpositive or rather non-zero."},{"Start":"02:27.810 ","End":"02:32.519","Text":"Um, non-negative result, so it could bezero as well, and that would be fine."},{"Start":"02:33.149 ","End":"02:38.760","Text":"So when we take the absolute value ofa positive number, well that\u0027s fine."},{"Start":"02:38.790 ","End":"02:41.700","Text":"We just leave it as it isbecause it\u0027s already greater"},{"Start":"02:41.700 ","End":"02:43.410","Text":"than or equal to zero, fine."},{"Start":"02:44.040 ","End":"02:45.450","Text":"When we have a negative number."},{"Start":"02:46.035 ","End":"02:48.765","Text":"Say X is minus four."},{"Start":"02:49.214 ","End":"02:51.114","Text":"Well, how do we make that positive?"},{"Start":"02:51.375 ","End":"02:55.304","Text":"Well, we just times that by a minusone, because minus times a minus"},{"Start":"02:55.304 ","End":"02:56.924","Text":"will return a positive number."},{"Start":"02:57.494 ","End":"03:03.494","Text":"And if you want to see this sort ofillustratively, well, if we draw a plot"},{"Start":"03:03.795 ","End":"03:12.404","Text":"and we have, you know, Exxon, the X axis,and we\u0027ve got, you know, Y on this axis,"},{"Start":"03:12.674 ","End":"03:14.714","Text":"and if we have the function, Y is equal."},{"Start":"03:16.210 ","End":"03:17.520","Text":"mods of X."},{"Start":"03:17.760 ","End":"03:23.310","Text":"Well, you know, that meansthat Y is equal to X when X is"},{"Start":"03:23.310 ","End":"03:24.690","Text":"greater than or equal to zero."},{"Start":"03:24.990 ","End":"03:28.580","Text":"So we just get kind of this thing here."},{"Start":"03:29.370 ","End":"03:29.970","Text":"Like that."},{"Start":"03:30.600 ","End":"03:37.170","Text":"And when X is less than zero,Y is equal to negative X."},{"Start":"03:37.380 ","End":"03:41.310","Text":"But remember, if X is is negativeand we\u0027re taking the negative"},{"Start":"03:41.310 ","End":"03:44.700","Text":"of that, then that\u0027s going toreturn a positive result for us."},{"Start":"03:44.700 ","End":"03:48.780","Text":"So we actually just getthis kind of nice symmetry."},{"Start":"03:49.965 ","End":"03:50.565","Text":"The graph."},{"Start":"03:51.075 ","End":"03:55.185","Text":"So that\u0027s sort of just to illustratethe point being made here."},{"Start":"03:55.665 ","End":"03:59.865","Text":"So modulars of X is equal tojust X when X is greater than"},{"Start":"03:59.865 ","End":"04:01.995","Text":"or equal to zero and negative x."},{"Start":"04:02.025 ","End":"04:03.465","Text":"When X is less than zero."},{"Start":"04:03.915 ","End":"04:08.625","Text":"Now I think this is more thepart where people might trip up."},{"Start":"04:08.895 ","End":"04:16.455","Text":"So if we consider component two andwe\u0027ve got the absolute value of X minus"},{"Start":"04:16.455 ","End":"04:19.125","Text":"two, well then how is this define."},{"Start":"04:19.844 ","End":"04:25.695","Text":"Well, this is just equalto X minus two and when?"},{"Start":"04:25.724 ","End":"04:29.635","Text":"When is it equal to X minus two, whenwhatever inside here is positive."},{"Start":"04:30.135 ","End":"04:38.234","Text":"So this is true when X minus two isgreater than or equal to zero, or in other"},{"Start":"04:38.240 ","End":"04:42.435","Text":"words, X is greater than or equal to two."},{"Start":"04:42.674 ","End":"04:46.664","Text":"If we just bring the two onthe other side and in the same"},{"Start":"04:46.664 ","End":"04:48.615","Text":"way we just take the negative."},{"Start":"04:48.945 ","End":"04:55.605","Text":"Of this for when X minus two isless than zero, which implies,"},{"Start":"04:55.635 ","End":"05:00.585","Text":"you know, X is um, less than two."},{"Start":"05:01.095 ","End":"05:05.805","Text":"So you can see already we\u0027re seeingthat these specific component parts"},{"Start":"05:05.805 ","End":"05:10.095","Text":"work in different ways, depend onthe domain that we\u0027re looking at."},{"Start":"05:10.665 ","End":"05:13.915","Text":"And actually the idea here is the."},{"Start":"05:14.594 ","End":"05:16.515","Text":"for when we\u0027re looking at X minus three."},{"Start":"05:16.515 ","End":"05:22.395","Text":"So we apply the exact same method,so now this is component three, the"},{"Start":"05:22.784 ","End":"05:28.544","Text":"absolute value of X minus three,and that\u0027s equal to X minus three."},{"Start":"05:28.875 ","End":"05:34.865","Text":"When X minus three is greater thanor equal to zero, which implies the"},{"Start":"05:34.870 ","End":"05:37.575","Text":"X is greater than or equal to three."},{"Start":"05:39.540 ","End":"05:44.640","Text":", it\u0027s the negative of this when Xis less than free, or let\u0027s just"},{"Start":"05:44.640 ","End":"05:45.870","Text":"do the steps in the normal way."},{"Start":"05:45.870 ","End":"05:49.830","Text":"We did when X minusthree is less than zero."},{"Start":"05:50.370 ","End":"05:53.220","Text":", which implies X is less than three."},{"Start":"05:53.490 ","End":"05:55.740","Text":"Now for speed."},{"Start":"05:55.740 ","End":"06:00.450","Text":"You can kind of just avoid thisstep here and just say, look, we"},{"Start":"06:00.450 ","End":"06:04.080","Text":"know that this is gonna be positivewhen X is bigger than free and"},{"Start":"06:04.110 ","End":"06:07.020","Text":"negative when X is less than free."},{"Start":"06:07.050 ","End":"06:11.969","Text":"But just for the sake of, you know,Having the rote method that might help"},{"Start":"06:11.969 ","End":"06:15.750","Text":"you for when you look at things that mightbe a bit more complicated, just if you"},{"Start":"06:15.750 ","End":"06:17.520","Text":"haven\u0027t done questions like this before."},{"Start":"06:18.000 ","End":"06:24.090","Text":"So now that we\u0027ve worked out how each partworks within specific domains, we just"},{"Start":"06:24.090 ","End":"06:26.070","Text":"now need to group all of this together."},{"Start":"06:26.520 ","End":"06:30.599","Text":"So basically we\u0027re gonna havethree different domains here."},{"Start":"06:30.960 ","End":"06:35.130","Text":"So we\u0027re going to have thedomain when X is less than zero."},{"Start":"06:35.640 ","End":"06:36.630","Text":"So that\u0027s one."},{"Start":"06:37.545 ","End":"06:39.405","Text":"X is less than zero."},{"Start":"06:40.035 ","End":"06:43.725","Text":"The other domain we\u0027ve got is whenX is greater than or equal to zero."},{"Start":"06:44.235 ","End":"06:48.345","Text":"So we\u0027ve got X is greater than orequal to zero, but less than two."},{"Start":"06:49.635 ","End":"06:50.055","Text":"Okay."},{"Start":"06:51.795 ","End":"06:53.205","Text":"And what other domains do we have?"},{"Start":"06:53.205 ","End":"06:57.675","Text":"Well, we\u0027ve got X is greaterthan or equal to zero or greater"},{"Start":"06:57.675 ","End":"06:58.995","Text":"than, or equal to two rather."},{"Start":"06:59.115 ","End":"07:03.415","Text":"So we\u0027ve got two hereand then less than three."},{"Start":"07:05.054 ","End":"07:08.625","Text":". And then the final domain isjust going to be when X is"},{"Start":"07:08.625 ","End":"07:12.135","Text":"greater than or equal to three."},{"Start":"07:13.065 ","End":"07:19.065","Text":"So just to reiterate wherethese sub-domains come from."},{"Start":"07:19.364 ","End":"07:24.765","Text":"Well, the X is less than zero, so let\u0027sjust call this I that comes from here."},{"Start":"07:26.054 ","End":"07:29.745","Text":"X is between zero andtwo, but not equal to two."},{"Start":"07:30.344 ","End":"07:31.155","Text":"Um, let\u0027s call."},{"Start":"07:32.205 ","End":"07:36.795","Text":"I I, and this comes from this parthere where X is bigger than or"},{"Start":"07:36.795 ","End":"07:40.425","Text":"equal to zero, but less than two."},{"Start":"07:41.415 ","End":"07:42.795","Text":"You can see where this is going."},{"Start":"07:42.855 ","End":"07:44.745","Text":"X is bigger than or equal to two."},{"Start":"07:45.135 ","End":"07:47.775","Text":"Let\u0027s call this number three."},{"Start":"07:47.865 ","End":"07:52.845","Text":"And that comes from this part,this part and this part here."},{"Start":"07:54.135 ","End":"07:56.145","Text":"And X is greater than or equal to free."},{"Start":"07:56.205 ","End":"07:56.805","Text":"Let\u0027s just call."},{"Start":"07:57.615 ","End":"08:00.465","Text":"Four IV and that will come from here."},{"Start":"08:00.975 ","End":"08:05.535","Text":"So now all we need to do is basicallyjust see how the function is"},{"Start":"08:05.535 ","End":"08:09.975","Text":"defined for each of these domainswithin this function, EF of X."},{"Start":"08:10.155 ","End":"08:12.735","Text":"So let\u0027s do the first one."},{"Start":"08:13.095 ","End":"08:18.585","Text":"So EF of X is equal, and thisis for X is less than zero."},{"Start":"08:20.609 ","End":"08:25.049","Text":"that\u0027s going to be, well, if X is lessthan zero, the mod X is going to be"},{"Start":"08:25.054 ","End":"08:27.570","Text":"equal to minus X as we defined it."},{"Start":"08:27.630 ","End":"08:34.919","Text":"So mod X is equal to minus x, andthen we\u0027ve got, um, so this next part"},{"Start":"08:34.919 ","End":"08:40.650","Text":"X minus two, well, if X is less thanzero, then it\u0027s certainly less than two."},{"Start":"08:40.980 ","End":"08:46.530","Text":"So this is going to be minus X minustwo, which is plus two minus x."},{"Start":"08:49.800 ","End":"08:54.839","Text":"And then similarly, we\u0027re going to havethis minus, and then in a bracket X minus"},{"Start":"08:54.839 ","End":"08:59.040","Text":"three, which is plus three, minus x."},{"Start":"08:59.670 ","End":"09:02.790","Text":"So if we\u0027re going to tidythis up, what do we have?"},{"Start":"09:03.150 ","End":"09:07.560","Text":"We\u0027ve got minus x minus X minusx, so that\u0027s minus three x"},{"Start":"09:07.800 ","End":"09:09.780","Text":"plus two, plus three plus five."},{"Start":"09:10.050 ","End":"09:15.329","Text":"So let\u0027s just tidy that up now, in whichcase we have minus three x plus five."},{"Start":"09:17.385 ","End":"09:24.015","Text":"minus three x plus five, andthat\u0027s for this domain here"},{"Start":"09:24.135 ","End":"09:26.235","Text":"where X is less than zero."},{"Start":"09:26.745 ","End":"09:32.745","Text":"Okay, now let\u0027s do X is greater thanor equal to zero, but less than two."},{"Start":"09:33.495 ","End":"09:37.575","Text":"So if that\u0027s true, well X isgreater than or equal to zero."},{"Start":"09:37.575 ","End":"09:38.835","Text":"So what\u0027s mod X?"},{"Start":"09:38.865 ","End":"09:40.545","Text":"That\u0027s just equal to X?"},{"Start":"09:40.845 ","End":"09:43.905","Text":"So we\u0027ve got, that\u0027s equal to x."},{"Start":"09:44.340 ","End":"09:51.180","Text":"Plus mod X minus two or X is still lessthan two, so we\u0027re defining that by this"},{"Start":"09:51.180 ","End":"09:58.200","Text":"plus two minus x, but again, which is theminus X minus two, so plus two minus x."},{"Start":"09:58.380 ","End":"10:02.220","Text":"And again, we\u0027ll be doing the minusX minus three here because two"},{"Start":"10:02.220 ","End":"10:04.110","Text":"is certainly smaller than three."},{"Start":"10:04.440 ","End":"10:07.830","Text":"So we\u0027re going to be using thenegative of this bit inside."},{"Start":"10:07.860 ","End":"10:10.860","Text":"So, Three minus X."},{"Start":"10:11.219 ","End":"10:14.370","Text":"So what do we get if we tidy this bit up?"},{"Start":"10:14.910 ","End":"10:18.930","Text":"Well, we\u0027ve got an X cancelingwith an X here, and then"},{"Start":"10:18.930 ","End":"10:20.969","Text":"we\u0027ve got a minus x plus five."},{"Start":"10:21.030 ","End":"10:26.880","Text":"So this is just goingto be minus x plus five."},{"Start":"10:27.975 ","End":"10:31.905","Text":", and I hope you can see thatactually these calculations just"},{"Start":"10:31.910 ","End":"10:33.555","Text":"become relatively straightforward."},{"Start":"10:33.585 ","End":"10:38.205","Text":"We just need to know what, whichpart of this definition to use"},{"Start":"10:38.415 ","End":"10:42.615","Text":"within both specific domains thatwe defined earlier in the question."},{"Start":"10:42.885 ","End":"10:49.605","Text":"So if X is greater than or equalto two, but less than three, well,"},{"Start":"10:49.785 ","End":"10:52.695","Text":"we know X is bigger than or equalto zero, so we\u0027re going to be using"},{"Start":"10:52.695 ","End":"10:56.505","Text":"the X part from component one plus."},{"Start":"10:56.910 ","End":"11:01.320","Text":"Now we know that X is greater than orequal to two, so we\u0027re going to be using"},{"Start":"11:01.320 ","End":"11:06.090","Text":"this first part of component two, sowe\u0027re going to be doing plus X minus two."},{"Start":"11:06.810 ","End":"11:11.130","Text":", but we\u0027re still less than three, sowe need to use this negative part of"},{"Start":"11:11.130 ","End":"11:14.459","Text":"here, which is plus three, minus x."},{"Start":"11:14.729 ","End":"11:19.890","Text":"So again, if we tidy this part upas well, then we\u0027ve got two x minus"},{"Start":"11:19.890 ","End":"11:24.270","Text":"x, which is a plus X minus twoplus three, which is a plus one."},{"Start":"11:24.599 ","End":"11:27.089","Text":"So here we just get X plus one."},{"Start":"11:27.390 ","End":"11:28.920","Text":"So let\u0027s write that down now."},{"Start":"11:29.280 ","End":"11:34.650","Text":"So x plus one, and thefinal part, I will let you."},{"Start":"11:35.370 ","End":"11:43.680","Text":"Yourself, and that\u0027s going to give youthree x minus five for this domain here,"},{"Start":"11:43.680 ","End":"11:45.840","Text":"where X is greater than or equal to three."},{"Start":"11:46.200 ","End":"11:50.910","Text":"So ultimately what we have is we havethe positive parts of all of these,"},{"Start":"11:50.940 ","End":"11:54.720","Text":"because if X is greater than or equalto three, it\u0027s certainly bigger than"},{"Start":"11:54.720 ","End":"12:00.300","Text":"or equal to two, and almost alsocertainly bigger than X is equal to zero."},{"Start":"12:00.690 ","End":"12:01.080","Text":"Okay?"},{"Start":"12:01.350 ","End":"12:03.350","Text":"So this is our solution."},{"Start":"12:03.355 ","End":"12:03.930","Text":"We don\u0027t need to."},{"Start":"12:05.040 ","End":"12:06.360","Text":", all of these equals here."},{"Start":"12:06.689 ","End":"12:09.510","Text":"Generally how we writethis is in this form."},{"Start":"12:09.510 ","End":"12:15.060","Text":"So we do these kind of big bracesand we say X is equal to this"},{"Start":"12:15.060 ","End":"12:18.180","Text":"part here, um, for this domain."},{"Start":"12:18.185 ","End":"12:20.160","Text":"So let\u0027s just bring this over here."},{"Start":"12:21.420 ","End":"12:26.069","Text":"And we basically just write itin a, in a piece wise manner."},{"Start":"12:27.105 ","End":"12:31.555","Text":"P by piece wise, we mean we\u0027vegot different pieces for specific"},{"Start":"12:31.560 ","End":"12:36.945","Text":"domains, rather than having it asthis kind of absolute value function."},{"Start":"12:37.305 ","End":"12:40.905","Text":"And you might be thinking, well, whydo we, why do we write it in this way?"},{"Start":"12:41.265 ","End":"12:45.975","Text":"And the reason for that is if we go tosketch this now, it\u0027s much simpler to"},{"Start":"12:46.215 ","End":"12:50.415","Text":"to sketch these straight line graphsthan trying to figure out what on"},{"Start":"12:50.420 ","End":"12:53.475","Text":"earth we\u0027re doing with all these sortsof different absolute value signs."},{"Start":"12:53.505 ","End":"12:54.795","Text":"So that is the motivation."},{"Start":"12:56.475 ","End":"12:57.015","Text":"Thank you."}],"ID":33898}],"Thumbnail":null,"ID":1186},{"Name":"The Absolute Value Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Absolute Value Function","Duration":"1m 16s","ChapterTopicVideoID":8272,"CourseChapterTopicPlaylistID":1187,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"Here we are with y equals absolute value of x,"},{"Start":"00:02.745 ","End":"00:04.635","Text":"very famous function,"},{"Start":"00:04.635 ","End":"00:07.155","Text":"1 you should certainly know about."},{"Start":"00:07.155 ","End":"00:08.640","Text":"Let\u0027s just do it with a table."},{"Start":"00:08.640 ","End":"00:10.050","Text":"Let\u0027s start right away."},{"Start":"00:10.050 ","End":"00:12.990","Text":"Now, notice that x can be anything,"},{"Start":"00:12.990 ","End":"00:14.910","Text":"there\u0027s no restriction on the domain."},{"Start":"00:14.910 ","End":"00:18.645","Text":"Domain is all x because whether x is positive or negative,"},{"Start":"00:18.645 ","End":"00:20.970","Text":"absolute value of x is defined,"},{"Start":"00:20.970 ","End":"00:22.740","Text":"but y will always be positive."},{"Start":"00:22.740 ","End":"00:25.455","Text":"Which is why I drew this piece a bit shorter because I know"},{"Start":"00:25.455 ","End":"00:28.620","Text":"everything is going to be positive or at least non-negative."},{"Start":"00:28.620 ","End":"00:32.040","Text":"When x is 0, the absolute value is defined as 0."},{"Start":"00:32.040 ","End":"00:35.085","Text":"Then if we take 1 or minus 1,"},{"Start":"00:35.085 ","End":"00:37.040","Text":"the absolute value is going to be 1."},{"Start":"00:37.040 ","End":"00:39.635","Text":"If we take 2 or minus 2,"},{"Start":"00:39.635 ","End":"00:42.400","Text":"the absolute value is going to be 2, and so on."},{"Start":"00:42.400 ","End":"00:45.175","Text":"We have 3 or minus 3,"},{"Start":"00:45.175 ","End":"00:47.435","Text":"then the absolute value is going to be 3."},{"Start":"00:47.435 ","End":"00:49.190","Text":"If I plot all these points,"},{"Start":"00:49.190 ","End":"00:52.105","Text":"0,0 is here, 1,1 would be here,"},{"Start":"00:52.105 ","End":"00:55.610","Text":"2,2 is here, 3,3 is here,"},{"Start":"00:55.610 ","End":"00:57.560","Text":"likewise on the minus 1."},{"Start":"00:57.560 ","End":"01:01.040","Text":"This is also 1, 2, 3,"},{"Start":"01:01.040 ","End":"01:04.240","Text":"minus 1 is also 1 minus 2,"},{"Start":"01:04.240 ","End":"01:07.055","Text":"2, minus 3, 3."},{"Start":"01:07.055 ","End":"01:10.430","Text":"It\u0027s supposed to come out a straight line. This is what it will look like."},{"Start":"01:10.430 ","End":"01:13.370","Text":"Of course, it will go on indefinitely to infinity."},{"Start":"01:13.370 ","End":"01:17.070","Text":"We made the introduction to absolute value of x."}],"ID":8447},{"Watched":false,"Name":"Exercise 1","Duration":"4m ","ChapterTopicVideoID":6141,"CourseChapterTopicPlaylistID":1187,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this exercise, we have to sketch the graph of the function f of x equals 1,"},{"Start":"00:04.980 ","End":"00:07.815","Text":"plus absolute value of x minus 2."},{"Start":"00:07.815 ","End":"00:12.030","Text":"Let\u0027s write the definition of absolute value of x minus 2,"},{"Start":"00:12.030 ","End":"00:15.060","Text":"and this will give us a piece-wise definition of the graph."},{"Start":"00:15.060 ","End":"00:18.765","Text":"Now, remember, just reminding you that in general,"},{"Start":"00:18.765 ","End":"00:25.335","Text":"the absolute value of x is defined to be equal to x,"},{"Start":"00:25.335 ","End":"00:32.620","Text":"for x bigger or equal to 0 and minus x when x is less than 0."},{"Start":"00:32.620 ","End":"00:36.025","Text":"In our case, we have x minus 2,"},{"Start":"00:36.025 ","End":"00:42.600","Text":"the absolute value of x minus 2 is equal to x minus 2,"},{"Start":"00:42.600 ","End":"00:47.225","Text":"when x minus 2 is bigger or equal to 0,"},{"Start":"00:47.225 ","End":"00:54.020","Text":"and to minus x minus 2 when x minus 2 is less than 0."},{"Start":"00:54.020 ","End":"00:56.240","Text":"Let\u0027s simplify this a bit."},{"Start":"00:56.240 ","End":"01:05.305","Text":"We get the absolute value of x minus 2 is equal to it\u0027s x minus 2,"},{"Start":"01:05.305 ","End":"01:08.825","Text":"when x minus 2 is bigger or equal to 0."},{"Start":"01:08.825 ","End":"01:15.950","Text":"In other words, when x is bigger or equal to 2 and minus x minus 2,"},{"Start":"01:15.950 ","End":"01:18.530","Text":"which is 2 minus x."},{"Start":"01:18.530 ","End":"01:21.230","Text":"Whenever x minus 2 less than 0,"},{"Start":"01:21.230 ","End":"01:24.155","Text":"which means that x is less than 2."},{"Start":"01:24.155 ","End":"01:26.750","Text":"Now, that\u0027s just part of f of x."},{"Start":"01:26.750 ","End":"01:28.325","Text":"If we write it out,"},{"Start":"01:28.325 ","End":"01:35.225","Text":"we\u0027ll get that f of x is equal to piece-wise."},{"Start":"01:35.225 ","End":"01:38.780","Text":"We just take what\u0027s above and add the 1 plus."},{"Start":"01:38.780 ","End":"01:48.695","Text":"X minus 2 plus 1 is x minus 1 and that\u0027s true when x is bigger or equal to 2."},{"Start":"01:48.695 ","End":"01:51.095","Text":"2 minus x plus 1,"},{"Start":"01:51.095 ","End":"01:57.800","Text":"which is 3 minus x when x is less than 2."},{"Start":"01:57.800 ","End":"02:00.110","Text":"Okay? What we have is a function f,"},{"Start":"02:00.110 ","End":"02:03.410","Text":"which is defined piecewise in 2 bits and in each piece,"},{"Start":"02:03.410 ","End":"02:05.225","Text":"it\u0027s a linear graph."},{"Start":"02:05.225 ","End":"02:07.480","Text":"Let\u0027s start drawing."},{"Start":"02:07.480 ","End":"02:11.780","Text":"That\u0027s the y-axis, that\u0027s the x-axis."},{"Start":"02:11.780 ","End":"02:16.475","Text":"The most important point on the x-axis is the point x equals 2."},{"Start":"02:16.475 ","End":"02:20.330","Text":"That say that somewhere here and now"},{"Start":"02:20.330 ","End":"02:24.245","Text":"what we\u0027ll do is draw each of these lines separately."},{"Start":"02:24.245 ","End":"02:29.330","Text":"Let\u0027s say that we draw the first 1 in turquoise."},{"Start":"02:29.330 ","End":"02:31.130","Text":"F of x is x minus 1."},{"Start":"02:31.130 ","End":"02:32.420","Text":"There\u0027s many ways to do this."},{"Start":"02:32.420 ","End":"02:34.925","Text":"1 way is to substitute values."},{"Start":"02:34.925 ","End":"02:37.300","Text":"For example, when x is 2,"},{"Start":"02:37.300 ","End":"02:41.960","Text":"why is 1 and when x is 0,"},{"Start":"02:41.960 ","End":"02:44.220","Text":"y is minus 1."},{"Start":"02:44.220 ","End":"02:51.089","Text":"We get something like this and this is the point 2,"},{"Start":"02:51.089 ","End":"02:54.605","Text":"1, the other bit 3 minus x,"},{"Start":"02:54.605 ","End":"02:56.240","Text":"use a different color."},{"Start":"02:56.240 ","End":"02:57.830","Text":"Also we can substitute,"},{"Start":"02:57.830 ","End":"02:59.270","Text":"let\u0027s say when x is 0,"},{"Start":"02:59.270 ","End":"03:02.180","Text":"we\u0027ll get y is 3."},{"Start":"03:02.180 ","End":"03:07.639","Text":"And also we could take x equals 2 and then y would equal 1 also."},{"Start":"03:07.639 ","End":"03:12.785","Text":"Actually passes through the same point and we get something like this."},{"Start":"03:12.785 ","End":"03:18.454","Text":"This is the graph of 3 minus x,"},{"Start":"03:18.454 ","End":"03:27.340","Text":"say y equals 3 minus x and the other 1 was y equals x minus 1."},{"Start":"03:27.340 ","End":"03:31.940","Text":"Now what we have to do is to take into the account the domains."},{"Start":"03:31.940 ","End":"03:34.835","Text":"When x is bigger or equal to 2,"},{"Start":"03:34.835 ","End":"03:39.480","Text":"we\u0027re on the blue graph from here to here,"},{"Start":"03:39.480 ","End":"03:45.635","Text":"and then going on to infinity and the green bit when x is less than 2,"},{"Start":"03:45.635 ","End":"03:47.915","Text":"which does not include this point,"},{"Start":"03:47.915 ","End":"03:51.290","Text":"but this point is here anyway on the blue graph,"},{"Start":"03:51.290 ","End":"03:53.130","Text":"so we just continue,"},{"Start":"03:53.130 ","End":"04:01.170","Text":"and the answer to the question is this black part of the graph and we\u0027re done."}],"ID":6153},{"Watched":false,"Name":"Exercise 2","Duration":"5m 17s","ChapterTopicVideoID":6142,"CourseChapterTopicPlaylistID":1187,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to graph the function x"},{"Start":"00:03.270 ","End":"00:07.890","Text":"squared plus twice absolute value of x plus 1 plus 1."},{"Start":"00:07.890 ","End":"00:11.625","Text":"I\u0027d like to remind you what the absolute value means."},{"Start":"00:11.625 ","End":"00:15.990","Text":"In general, the absolute value of some number a is"},{"Start":"00:15.990 ","End":"00:21.539","Text":"equal to a itself if a is bigger or equal to 0,"},{"Start":"00:21.539 ","End":"00:26.250","Text":"and minus a if a is less than 0."},{"Start":"00:26.250 ","End":"00:29.910","Text":"In our case, the a is x plus 1."},{"Start":"00:29.910 ","End":"00:40.230","Text":"This is a piece-wise function and we can write it as f of x equals 2 pieces x"},{"Start":"00:40.230 ","End":"00:46.300","Text":"squared plus twice x plus 1 itself plus 1 in"},{"Start":"00:46.300 ","End":"00:53.615","Text":"the case that a bigger equal to 0 means x plus 1 bigger or equal to 0."},{"Start":"00:53.615 ","End":"00:57.935","Text":"The other piece is x squared, from the minus a,"},{"Start":"00:57.935 ","End":"01:02.980","Text":"we put minus twice x plus 1, plus 1."},{"Start":"01:02.980 ","End":"01:07.325","Text":"That\u0027s for the case where x plus 1 is less than 0."},{"Start":"01:07.325 ","End":"01:13.385","Text":"Let\u0027s simplify this a bit and write f of x equals."},{"Start":"01:13.385 ","End":"01:15.949","Text":"Just expanding this with simple algebra,"},{"Start":"01:15.949 ","End":"01:22.195","Text":"we get x squared plus 2x plus 3."},{"Start":"01:22.195 ","End":"01:24.700","Text":"In the case x plus 1 bigger or equal to 0,"},{"Start":"01:24.700 ","End":"01:29.080","Text":"which just means that x is bigger or equal to minus 1."},{"Start":"01:29.080 ","End":"01:34.530","Text":"Expanding this, we get x squared minus 2x minus 2 plus"},{"Start":"01:34.530 ","End":"01:40.390","Text":"1 minus 1 for the case where x is less than minus 1."},{"Start":"01:40.390 ","End":"01:43.810","Text":"Each of these 2 pieces is a parabola,"},{"Start":"01:43.810 ","End":"01:45.790","Text":"and we\u0027ll sketch each one separately."},{"Start":"01:45.790 ","End":"01:48.350","Text":"First, let\u0027s draw some axes."},{"Start":"01:48.350 ","End":"01:50.725","Text":"Let\u0027s look at the first parabola,"},{"Start":"01:50.725 ","End":"01:53.185","Text":"x squared plus 2x plus 3."},{"Start":"01:53.185 ","End":"01:57.235","Text":"To plot it, we need a couple of points to a 3."},{"Start":"01:57.235 ","End":"01:59.440","Text":"Usually, we take the apex,"},{"Start":"01:59.440 ","End":"02:02.610","Text":"which is where x is minus b over 2a."},{"Start":"02:02.610 ","End":"02:05.930","Text":"In this case, x equals minus 2 over 2,"},{"Start":"02:05.930 ","End":"02:07.490","Text":"which is minus 1."},{"Start":"02:07.490 ","End":"02:09.700","Text":"When x is minus 1,"},{"Start":"02:09.700 ","End":"02:11.655","Text":"that\u0027s one of the points we take,"},{"Start":"02:11.655 ","End":"02:17.300","Text":"we get that y is equal to minus 1 squared,"},{"Start":"02:17.300 ","End":"02:21.065","Text":"which is 1 minus 2 plus 3."},{"Start":"02:21.065 ","End":"02:22.820","Text":"That gives us 2."},{"Start":"02:22.820 ","End":"02:26.365","Text":"In other words, we want to plot the point minus 1, 2."},{"Start":"02:26.365 ","End":"02:30.520","Text":"Easiest thing to do to get another point would be to put x equals 0,"},{"Start":"02:30.520 ","End":"02:32.710","Text":"the intersection with the y axis,"},{"Start":"02:32.710 ","End":"02:36.230","Text":"and when x is 0, we get y equals 3."},{"Start":"02:36.230 ","End":"02:39.080","Text":"In other words, we plot the point 0,"},{"Start":"02:39.080 ","End":"02:43.025","Text":"3, and let\u0027s do that in 1 color."},{"Start":"02:43.025 ","End":"02:45.375","Text":"Minus 1, 2,"},{"Start":"02:45.375 ","End":"02:49.759","Text":"somewhere here, and 0, 3 here."},{"Start":"02:49.759 ","End":"02:52.640","Text":"This is the apex, so it\u0027s symmetrical about this."},{"Start":"02:52.640 ","End":"02:56.540","Text":"This is just a rough sketch, something like this."},{"Start":"02:56.540 ","End":"03:01.220","Text":"However, we don\u0027t want the complete parabola because the top part"},{"Start":"03:01.220 ","End":"03:05.975","Text":"only applies when x is bigger or equal to minus 1,"},{"Start":"03:05.975 ","End":"03:08.375","Text":"which is from minus 1 onward."},{"Start":"03:08.375 ","End":"03:10.940","Text":"In other words, what we need to do,"},{"Start":"03:10.940 ","End":"03:13.834","Text":"is to erase the parts we don\u0027t want."},{"Start":"03:13.834 ","End":"03:16.850","Text":"There we go. Tidied up the graph a bit."},{"Start":"03:16.850 ","End":"03:18.380","Text":"It wouldn\u0027t look so good."},{"Start":"03:18.380 ","End":"03:20.555","Text":"Let\u0027s continue to the second half,"},{"Start":"03:20.555 ","End":"03:23.015","Text":"the x squared minus 2x minus 1,"},{"Start":"03:23.015 ","End":"03:24.979","Text":"and plot a few points."},{"Start":"03:24.979 ","End":"03:26.900","Text":"I always like to take the apex,"},{"Start":"03:26.900 ","End":"03:28.670","Text":"which is minus b over 2a,"},{"Start":"03:28.670 ","End":"03:32.125","Text":"in this case, x equals 1."},{"Start":"03:32.125 ","End":"03:34.700","Text":"That gives us, if we compute in our heads,"},{"Start":"03:34.700 ","End":"03:38.390","Text":"y is 1 squared minus 2 minus 1,"},{"Start":"03:38.390 ","End":"03:41.035","Text":"and that gives us minus 2."},{"Start":"03:41.035 ","End":"03:42.270","Text":"In other words,"},{"Start":"03:42.270 ","End":"03:45.545","Text":"we want to plot the point 1 minus 2."},{"Start":"03:45.545 ","End":"03:50.730","Text":"Another easy point to plot is the interception with the y-axis"},{"Start":"03:50.730 ","End":"03:56.060","Text":"where x equals 0 and we immediately see that that gives us y equals minus 1,"},{"Start":"03:56.060 ","End":"03:59.705","Text":"so we have another, 0, minus 1."},{"Start":"03:59.705 ","End":"04:01.954","Text":"But in the case of a piece-wise function,"},{"Start":"04:01.954 ","End":"04:07.860","Text":"I also like to take the value of the border between 2 domains."},{"Start":"04:09.340 ","End":"04:12.590","Text":"In the previous case, I already had minus 1,"},{"Start":"04:12.590 ","End":"04:14.045","Text":"but let\u0027s take it here."},{"Start":"04:14.045 ","End":"04:16.180","Text":"X equals minus 1,"},{"Start":"04:16.180 ","End":"04:19.880","Text":"and substitute, and I\u0027ll just give you the answer."},{"Start":"04:19.880 ","End":"04:21.050","Text":"Y comes out too,"},{"Start":"04:21.050 ","End":"04:22.580","Text":"you can do the computation."},{"Start":"04:22.580 ","End":"04:25.630","Text":"We get minus 1, 2."},{"Start":"04:25.630 ","End":"04:29.480","Text":"Let\u0027s plot these points and the graph in a different color."},{"Start":"04:29.480 ","End":"04:31.545","Text":"Let\u0027s choose blue this time."},{"Start":"04:31.545 ","End":"04:34.850","Text":"I drew it for you and didn\u0027t come out the greatest."},{"Start":"04:34.850 ","End":"04:39.245","Text":"That\u0027s just a sketch. Here\u0027s the point 1 minus 2."},{"Start":"04:39.245 ","End":"04:41.850","Text":"Here\u0027s the point 0 minus 1."},{"Start":"04:41.850 ","End":"04:43.365","Text":"The minus 1,"},{"Start":"04:43.365 ","End":"04:44.900","Text":"2 comes out here,"},{"Start":"04:44.900 ","End":"04:48.230","Text":"which is exactly the same as the green point we had there."},{"Start":"04:48.230 ","End":"04:51.505","Text":"That was the apex. It\u0027s symmetrical here."},{"Start":"04:51.505 ","End":"04:54.855","Text":"Just draw a line through this."},{"Start":"04:54.855 ","End":"05:01.330","Text":"Again, we have to remember that this is defined for x less than minus 1."},{"Start":"05:01.330 ","End":"05:05.285","Text":"We have to only take the part from minus 1 and to the left,"},{"Start":"05:05.285 ","End":"05:07.850","Text":"which means we need to erase all the bits"},{"Start":"05:07.850 ","End":"05:11.225","Text":"from minus 1 to the right, and I just did that."},{"Start":"05:11.225 ","End":"05:13.400","Text":"This is a rough sketch of the answer."},{"Start":"05:13.400 ","End":"05:14.450","Text":"Could have been drawn better,"},{"Start":"05:14.450 ","End":"05:18.160","Text":"but that\u0027s the general idea, and we\u0027re done."}],"ID":6154},{"Watched":false,"Name":"Exercise 3","Duration":"3m 24s","ChapterTopicVideoID":6143,"CourseChapterTopicPlaylistID":1187,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to graph the function f of"},{"Start":"00:03.060 ","End":"00:07.590","Text":"x equals the absolute value of x squared plus x minus 2."},{"Start":"00:07.590 ","End":"00:10.545","Text":"I want to remind you what absolute value means."},{"Start":"00:10.545 ","End":"00:14.310","Text":"It\u0027s defined piece wise that the absolute value of some number,"},{"Start":"00:14.310 ","End":"00:21.135","Text":"let\u0027s say a is equal to a whenever a is bigger or equal to 0,"},{"Start":"00:21.135 ","End":"00:26.070","Text":"and minus a whenever a is less than 0."},{"Start":"00:26.070 ","End":"00:30.165","Text":"In our case, if we write the function in piece wise form,"},{"Start":"00:30.165 ","End":"00:32.820","Text":"we get f of x is equal to,"},{"Start":"00:32.820 ","End":"00:35.910","Text":"here a is this whole x squared plus x minus 2."},{"Start":"00:35.910 ","End":"00:40.905","Text":"It\u0027s equal to the x squared plus x minus 2 itself."},{"Start":"00:40.905 ","End":"00:47.540","Text":"When x squared plus x minus 2 is bigger or equal to 0,"},{"Start":"00:47.540 ","End":"00:50.000","Text":"but minus the same thing."},{"Start":"00:50.000 ","End":"00:55.820","Text":"Minus x squared plus x minus 2 whenever this is less than 0."},{"Start":"00:55.820 ","End":"01:02.695","Text":"In other words, x squared plus x minus 2 is less than 0."},{"Start":"01:02.695 ","End":"01:06.445","Text":"What we have to do is solve 2 inequalities,"},{"Start":"01:06.445 ","End":"01:07.850","Text":"x squared plus x minus 2,"},{"Start":"01:07.850 ","End":"01:11.035","Text":"bigger or equal to 0 or less than 0."},{"Start":"01:11.035 ","End":"01:15.050","Text":"The way we do it is by solving the quadratic equation,"},{"Start":"01:15.050 ","End":"01:23.625","Text":"x squared plus x minus 2 equals 0 and using the roots to help us solve the inequality."},{"Start":"01:23.625 ","End":"01:25.750","Text":"Now, I\u0027m assuming you know how to solve"},{"Start":"01:25.750 ","End":"01:28.345","Text":"quadratic equations and I won\u0027t do all the work here."},{"Start":"01:28.345 ","End":"01:33.910","Text":"I\u0027ll come straight to the answer that x is equal to either 1 or minus 2."},{"Start":"01:33.910 ","End":"01:35.805","Text":"Okay, let\u0027s start sketching."},{"Start":"01:35.805 ","End":"01:38.290","Text":"It\u0027s piece wise, so we\u0027ll do each piece separately."},{"Start":"01:38.290 ","End":"01:41.065","Text":"The top 1 is x squared plus x minus 2."},{"Start":"01:41.065 ","End":"01:42.730","Text":"It\u0027s a quadratic function,"},{"Start":"01:42.730 ","End":"01:46.720","Text":"so it\u0027s a parabola and the coefficient of x squared is positive,"},{"Start":"01:46.720 ","End":"01:49.405","Text":"so we have an upward facing parabola."},{"Start":"01:49.405 ","End":"01:53.635","Text":"We also know the roots are 1 and minus 2."},{"Start":"01:53.635 ","End":"01:56.200","Text":"We can do a rough sketch,"},{"Start":"01:56.200 ","End":"01:59.525","Text":"1 minus 2,"},{"Start":"01:59.525 ","End":"02:04.075","Text":"continue here and continue on from here."},{"Start":"02:04.075 ","End":"02:05.790","Text":"Something like this."},{"Start":"02:05.790 ","End":"02:07.770","Text":"That\u0027s the complete parabola,"},{"Start":"02:07.770 ","End":"02:10.055","Text":"but we have to confine ourselves to"},{"Start":"02:10.055 ","End":"02:15.605","Text":"where it\u0027s defined x less than or equal to minus 2 or x bigger or equal to 1."},{"Start":"02:15.605 ","End":"02:20.539","Text":"Other words from minus 2 and in the downward direction,"},{"Start":"02:20.539 ","End":"02:22.055","Text":"or from 1,"},{"Start":"02:22.055 ","End":"02:25.254","Text":"then the rightward direction."},{"Start":"02:25.254 ","End":"02:28.745","Text":"It means we have to just erase the middle bit."},{"Start":"02:28.745 ","End":"02:31.415","Text":"Now for the bottom part,"},{"Start":"02:31.415 ","End":"02:33.785","Text":"this time it\u0027s also a parabola,"},{"Start":"02:33.785 ","End":"02:35.630","Text":"but it\u0027s downward facing because"},{"Start":"02:35.630 ","End":"02:38.869","Text":"the coefficient of x squared is minus 1, which is negative."},{"Start":"02:38.869 ","End":"02:41.300","Text":"It\u0027s a downward facing parabola,"},{"Start":"02:41.300 ","End":"02:43.390","Text":"and it also has the same roots."},{"Start":"02:43.390 ","End":"02:44.990","Text":"Because if we make this equal to 0,"},{"Start":"02:44.990 ","End":"02:47.360","Text":"it\u0027s the same as making its negative equal to 0."},{"Start":"02:47.360 ","End":"02:51.290","Text":"We need a downward facing parabola that goes through 1 and minus 2."},{"Start":"02:51.290 ","End":"02:53.780","Text":"That\u0027s the same points as we had before."},{"Start":"02:53.780 ","End":"02:57.475","Text":"The complete parabola would look something,"},{"Start":"02:57.475 ","End":"03:01.250","Text":"and this time, we also have to look at the domain."},{"Start":"03:01.250 ","End":"03:06.830","Text":"It\u0027s only defined this way between minus 2 and 1, not including."},{"Start":"03:06.830 ","End":"03:08.975","Text":"In other words, the bit from,"},{"Start":"03:08.975 ","End":"03:11.300","Text":"from here to here,"},{"Start":"03:11.300 ","End":"03:14.015","Text":"along here, but not the rest of it."},{"Start":"03:14.015 ","End":"03:17.650","Text":"We need to erase this part and this part,"},{"Start":"03:17.650 ","End":"03:20.780","Text":"which gives us this graph, could be prettier,"},{"Start":"03:20.780 ","End":"03:25.080","Text":"but you get the general idea. We\u0027re done."}],"ID":6155}],"Thumbnail":null,"ID":1187}]

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1.1

1