[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.20,0:00:06.74,Default,,0000,0000,0000,,The composition of two functions\NG&F is the function we get from
Dialogue: 0,0:00:06.74,0:00:11.65,Default,,0000,0000,0000,,doing at 1st and then G. So if\Nwe let F of X.
Dialogue: 0,0:00:13.47,0:00:16.58,Default,,0000,0000,0000,,Be given by F of X\Nequals X squared.
Dialogue: 0,0:00:17.62,0:00:19.12,Default,,0000,0000,0000,,And we let G of X.
Dialogue: 0,0:00:20.78,0:00:22.67,Default,,0000,0000,0000,,BX +3.
Dialogue: 0,0:00:24.32,0:00:30.36,Default,,0000,0000,0000,,Then GF of X is given by.\NWell, we do F of X first and
Dialogue: 0,0:00:30.36,0:00:31.57,Default,,0000,0000,0000,,that's X squared.
Dialogue: 0,0:00:32.94,0:00:34.96,Default,,0000,0000,0000,,Then we apply G to the\Nwhole of that.
Dialogue: 0,0:00:38.23,0:00:40.68,Default,,0000,0000,0000,,Now, since G just adds 3 onto a
Dialogue: 0,0:00:40.68,0:00:45.19,Default,,0000,0000,0000,,number. G of X squared is\NX squared +3.
Dialogue: 0,0:00:46.43,0:00:49.17,Default,,0000,0000,0000,,So GF of X is X squared +3.
Dialogue: 0,0:00:50.89,0:00:52.09,Default,,0000,0000,0000,,Here's another example.
Dialogue: 0,0:00:54.61,0:00:56.44,Default,,0000,0000,0000,,This time will let F of X.
Dialogue: 0,0:00:57.05,0:00:59.91,Default,,0000,0000,0000,,Be given by F of X equals 2X.
Dialogue: 0,0:01:01.31,0:01:02.66,Default,,0000,0000,0000,,And we'll have G of X.
Dialogue: 0,0:01:04.12,0:01:05.53,Default,,0000,0000,0000,,Being East The X.
Dialogue: 0,0:01:07.71,0:01:12.94,Default,,0000,0000,0000,,Now to get GFX again, we write\Ndown what F of X is.
Dialogue: 0,0:01:13.87,0:01:15.07,Default,,0000,0000,0000,,That was 2X.
Dialogue: 0,0:01:17.10,0:01:19.31,Default,,0000,0000,0000,,And then we apply G\Nto the whole of that.
Dialogue: 0,0:01:22.30,0:01:24.60,Default,,0000,0000,0000,,Now G of X is E to the X.
Dialogue: 0,0:01:25.16,0:01:28.50,Default,,0000,0000,0000,,So G of two X is E to the 2X.
Dialogue: 0,0:01:29.69,0:01:33.59,Default,,0000,0000,0000,,So again we have here GF of X is\NE to the 2X.
Dialogue: 0,0:01:35.06,0:01:38.12,Default,,0000,0000,0000,,Sometimes the composition of two\Nfunctions is called the function
Dialogue: 0,0:01:38.12,0:01:41.49,Default,,0000,0000,0000,,of a function, and here's\Nanother way of writing it down.
Dialogue: 0,0:01:42.77,0:01:49.63,Default,,0000,0000,0000,,We sometimes writes GF of\NX as G Little Circle F
Dialogue: 0,0:01:49.63,0:01:50.88,Default,,0000,0000,0000,,of X.
Dialogue: 0,0:01:52.75,0:01:55.76,Default,,0000,0000,0000,,Now you don't have to use this\Nnotation yourself, but you
Dialogue: 0,0:01:55.76,0:01:58.50,Default,,0000,0000,0000,,might see it in classes or\Nworkbooks, so it's important
Dialogue: 0,0:01:58.50,0:01:59.87,Default,,0000,0000,0000,,to know what it means.
Dialogue: 0,0:02:03.92,0:02:06.89,Default,,0000,0000,0000,,The order in which we compose\Nfunctions can make a big
Dialogue: 0,0:02:06.89,0:02:12.39,Default,,0000,0000,0000,,difference. If you remember our\Nfirst example, we had F of X
Dialogue: 0,0:02:12.39,0:02:13.63,Default,,0000,0000,0000,,being X squared.
Dialogue: 0,0:02:15.08,0:02:16.15,Default,,0000,0000,0000,,And G of X.
Dialogue: 0,0:02:17.53,0:02:18.92,Default,,0000,0000,0000,,Being X +3.
Dialogue: 0,0:02:21.04,0:02:26.89,Default,,0000,0000,0000,,We'll see what happens when we\Ndo G 1st and then F, so this is
Dialogue: 0,0:02:26.89,0:02:28.06,Default,,0000,0000,0000,,FG of X.
Dialogue: 0,0:02:29.41,0:02:33.19,Default,,0000,0000,0000,,So this time we write down what\NG of X is first, which is X +3.
Dialogue: 0,0:02:36.02,0:02:38.86,Default,,0000,0000,0000,,Then we apply F to\Nthe whole of that.
Dialogue: 0,0:02:40.41,0:02:45.77,Default,,0000,0000,0000,,Now applying the F to something\Njust makes it square, so F of X
Dialogue: 0,0:02:45.77,0:02:48.45,Default,,0000,0000,0000,,+3 is X plus three all squared.
Dialogue: 0,0:02:49.86,0:02:54.07,Default,,0000,0000,0000,,And that X squared plus\N6X plus 9.
Dialogue: 0,0:02:55.45,0:03:00.55,Default,,0000,0000,0000,,And if you remember, GF of X was\Njust X squared +3.
Dialogue: 0,0:03:02.44,0:03:05.14,Default,,0000,0000,0000,,So you can see here that\Nchanging the order in which
Dialogue: 0,0:03:05.14,0:03:07.10,Default,,0000,0000,0000,,we compose these functions\Nmakes a big difference.
Dialogue: 0,0:03:14.80,0:03:18.06,Default,,0000,0000,0000,,Now we've looked at how to\Ncompose functions. We look at
Dialogue: 0,0:03:18.06,0:03:19.24,Default,,0000,0000,0000,,how to decompose functions.
Dialogue: 0,0:03:20.74,0:03:26.47,Default,,0000,0000,0000,,If I have a function like each\Nof X equals E to the 2X.
Dialogue: 0,0:03:27.36,0:03:30.65,Default,,0000,0000,0000,,We've already seen that we\Ncan write H as a composition
Dialogue: 0,0:03:30.65,0:03:31.55,Default,,0000,0000,0000,,of two functions.
Dialogue: 0,0:03:33.13,0:03:40.98,Default,,0000,0000,0000,,H of X was equal to GF\Nof X, where F of X was
Dialogue: 0,0:03:40.98,0:03:46.59,Default,,0000,0000,0000,,2X and G of X was E\Nto the X.
Dialogue: 0,0:03:53.61,0:03:57.35,Default,,0000,0000,0000,,Let's decompose another\Nfunction. This time will have
Dialogue: 0,0:03:57.35,0:03:59.21,Default,,0000,0000,0000,,function each of X.
Dialogue: 0,0:04:00.48,0:04:01.86,Default,,0000,0000,0000,,Equals the cube roots.
Dialogue: 0,0:04:02.89,0:04:04.94,Default,,0000,0000,0000,,Of 2X plus one.
Dialogue: 0,0:04:07.33,0:04:14.83,Default,,0000,0000,0000,,The first here we send X to 2X\NPlus One and then we cube root
Dialogue: 0,0:04:14.83,0:04:22.83,Default,,0000,0000,0000,,the whole thing. So if we let F\Nof XB2X plus one then H of X.
Dialogue: 0,0:04:24.38,0:04:25.40,Default,,0000,0000,0000,,Is the cube root?
Dialogue: 0,0:04:26.44,0:04:27.51,Default,,0000,0000,0000,,Oh, this is X.
Dialogue: 0,0:04:29.36,0:04:31.49,Default,,0000,0000,0000,,So then if we sat G of X.
Dialogue: 0,0:04:32.66,0:04:34.15,Default,,0000,0000,0000,,To be cube root X.
Dialogue: 0,0:04:35.77,0:04:37.50,Default,,0000,0000,0000,,We get that HX.
Dialogue: 0,0:04:38.91,0:04:40.47,Default,,0000,0000,0000,,Is G.
Dialogue: 0,0:04:42.00,0:04:47.23,Default,,0000,0000,0000,,Of F of X. So that's\NGF of X.
Dialogue: 0,0:04:48.42,0:04:50.99,Default,,0000,0000,0000,,Being able to decompose\Nfunctions will be very
Dialogue: 0,0:04:50.99,0:04:54.20,Default,,0000,0000,0000,,important when you come to do\Nthings like integration and
Dialogue: 0,0:04:54.20,0:04:54.52,Default,,0000,0000,0000,,Differentiation.
Dialogue: 0,0:05:02.90,0:05:05.81,Default,,0000,0000,0000,,Now there is some pairs of\Nfunctions that can't be
Dialogue: 0,0:05:05.81,0:05:13.15,Default,,0000,0000,0000,,composed. Let's have F of\NX being minus X squared.
Dialogue: 0,0:05:14.00,0:05:15.55,Default,,0000,0000,0000,,And we'll have G of X.
Dialogue: 0,0:05:16.93,0:05:20.50,Default,,0000,0000,0000,,Being natural log of X.
Dialogue: 0,0:05:22.65,0:05:24.36,Default,,0000,0000,0000,,Now F of X.
Dialogue: 0,0:05:25.66,0:05:27.55,Default,,0000,0000,0000,,Is less than or equal to 0?
Dialogue: 0,0:05:28.27,0:05:29.94,Default,,0000,0000,0000,,Whichever X you pick.
Dialogue: 0,0:05:31.19,0:05:36.30,Default,,0000,0000,0000,,But G of X is only defined if X\Nis greater than 0.
Dialogue: 0,0:05:37.41,0:05:40.36,Default,,0000,0000,0000,,So G of F of X.
Dialogue: 0,0:05:41.55,0:05:42.82,Default,,0000,0000,0000,,Doesn't exist.
Dialogue: 0,0:05:53.75,0:05:56.32,Default,,0000,0000,0000,,There are also some pairs of\Nfunctions that can be
Dialogue: 0,0:05:56.32,0:05:58.63,Default,,0000,0000,0000,,composed with some ex, but\Nnot for other ex.
Dialogue: 0,0:05:59.72,0:06:06.46,Default,,0000,0000,0000,,This time will have F of\NX equals 4X minus 6.
Dialogue: 0,0:06:07.45,0:06:08.81,Default,,0000,0000,0000,,And we'll have geovax.
Dialogue: 0,0:06:11.16,0:06:13.65,Default,,0000,0000,0000,,Bing square root of X.
Dialogue: 0,0:06:15.29,0:06:20.54,Default,,0000,0000,0000,,Now G of X is only defined if X\Nis greater than or equal to 0.
Dialogue: 0,0:06:21.51,0:06:25.57,Default,,0000,0000,0000,,So GF of X can only be\Ndefined if F of X is greater
Dialogue: 0,0:06:25.57,0:06:27.02,Default,,0000,0000,0000,,than or equal to 0.
Dialogue: 0,0:06:28.66,0:06:33.63,Default,,0000,0000,0000,,And F of X is greater than or\Nequal to 0.
Dialogue: 0,0:06:34.35,0:06:38.39,Default,,0000,0000,0000,,Only if X is greater than or\Nequal to three over 2.
Dialogue: 0,0:06:42.14,0:06:46.61,Default,,0000,0000,0000,,So GF of X is only\Ndefined.
Dialogue: 0,0:06:50.32,0:06:54.74,Default,,0000,0000,0000,,4X greater than or equal to\Nthree over 2.
Dialogue: 0,0:06:57.50,0:07:01.36,Default,,0000,0000,0000,,The values of X for which\NGF is defined is called the
Dialogue: 0,0:07:01.36,0:07:02.33,Default,,0000,0000,0000,,domain of GF.
Dialogue: 0,0:07:08.95,0:07:12.34,Default,,0000,0000,0000,,The range of a function is the\Nset of all the values of
Dialogue: 0,0:07:12.34,0:07:16.66,Default,,0000,0000,0000,,function can take.\NSo for example, if F
Dialogue: 0,0:07:16.66,0:07:19.97,Default,,0000,0000,0000,,of X was each the X.
Dialogue: 0,0:07:21.63,0:07:22.59,Default,,0000,0000,0000,,The range.
Dialogue: 0,0:07:24.33,0:07:29.82,Default,,0000,0000,0000,,Is given by F of X is greater\Nthan zero because E to the X can
Dialogue: 0,0:07:29.82,0:07:31.53,Default,,0000,0000,0000,,only be greater than 0.
Dialogue: 0,0:07:32.59,0:07:37.55,Default,,0000,0000,0000,,Similarly, if we have F of X\Nbeing say, Sign X.
Dialogue: 0,0:07:40.72,0:07:41.74,Default,,0000,0000,0000,,Then the range.
Dialogue: 0,0:07:45.80,0:07:47.42,Default,,0000,0000,0000,,Is FX.
Dialogue: 0,0:07:48.88,0:07:52.93,Default,,0000,0000,0000,,Is between plus one and\Nminus one.
Dialogue: 0,0:08:00.46,0:08:03.31,Default,,0000,0000,0000,,The range of a composite\Nfunction must lie within the
Dialogue: 0,0:08:03.31,0:08:05.02,Default,,0000,0000,0000,,range of the second function we
Dialogue: 0,0:08:05.02,0:08:07.04,Default,,0000,0000,0000,,apply. Here's an example to show
Dialogue: 0,0:08:07.04,0:08:12.53,Default,,0000,0000,0000,,you this. Will put\Ndown F of X equals X
Dialogue: 0,0:08:12.53,0:08:15.62,Default,,0000,0000,0000,,minus 8 and G of X?
Dialogue: 0,0:08:16.99,0:08:18.27,Default,,0000,0000,0000,,Equals X squared.
Dialogue: 0,0:08:19.92,0:08:21.26,Default,,0000,0000,0000,,Now for F of X.
Dialogue: 0,0:08:21.86,0:08:22.78,Default,,0000,0000,0000,,The range.
Dialogue: 0,0:08:25.73,0:08:27.94,Default,,0000,0000,0000,,Is well F of X can take\Nany value.
Dialogue: 0,0:08:35.35,0:08:37.18,Default,,0000,0000,0000,,But for G of X.
Dialogue: 0,0:08:38.47,0:08:42.84,Default,,0000,0000,0000,,Ranges G of X is greater than or\Nequal to 0.
Dialogue: 0,0:08:44.42,0:08:47.51,Default,,0000,0000,0000,,Now let's look at the range of\Nthe composed function GF.
Dialogue: 0,0:08:49.19,0:08:55.14,Default,,0000,0000,0000,,GFX is well F of X\Nis X minus 8.
Dialogue: 0,0:08:57.67,0:09:05.01,Default,,0000,0000,0000,,And G of X squares it, so\NGF of X is X minus 8
Dialogue: 0,0:09:05.01,0:09:12.26,Default,,0000,0000,0000,,squared. Which is X\Nsquared minus 16X plus
Dialogue: 0,0:09:12.26,0:09:13.12,Default,,0000,0000,0000,,64.
Dialogue: 0,0:09:15.36,0:09:19.48,Default,,0000,0000,0000,,But since GF of X is\Nsomething squared.
Dialogue: 0,0:09:20.62,0:09:23.20,Default,,0000,0000,0000,,And all square numbers are\Ngreater than or equal to 0.
Dialogue: 0,0:09:24.05,0:09:27.18,Default,,0000,0000,0000,,GF of X is greater\Nthan or equal to 0.
Dialogue: 0,0:09:28.45,0:09:29.59,Default,,0000,0000,0000,,So the range here.
Dialogue: 0,0:09:32.32,0:09:36.69,Default,,0000,0000,0000,,Is GF of X is greater than or\Nequal to 0?
Dialogue: 0,0:09:38.26,0:09:43.15,Default,,0000,0000,0000,,So you can see here that the\Nrange of GF is the range of G.
Dialogue: 0,0:09:44.51,0:09:46.37,Default,,0000,0000,0000,,Now let's look at what\Nhappens when he composed
Dialogue: 0,0:09:46.37,0:09:47.41,Default,,0000,0000,0000,,and the other way around.
Dialogue: 0,0:09:48.78,0:09:50.98,Default,,0000,0000,0000,,So this time FG books.
Dialogue: 0,0:09:53.99,0:09:56.48,Default,,0000,0000,0000,,What again G of X is X squared?
Dialogue: 0,0:09:59.15,0:10:04.88,Default,,0000,0000,0000,,And applying eh? The whole of\Nthat gives us X squared minus 8.
Dialogue: 0,0:10:06.41,0:10:09.41,Default,,0000,0000,0000,,Now, since G of X is greater\Nthan or equal to 0.
Dialogue: 0,0:10:10.10,0:10:11.79,Default,,0000,0000,0000,,And here we're taking away 8.
Dialogue: 0,0:10:12.59,0:10:16.79,Default,,0000,0000,0000,,This whole thing must be greater\Nthan or equal to minus 8.
Dialogue: 0,0:10:18.82,0:10:19.95,Default,,0000,0000,0000,,So the range.
Dialogue: 0,0:10:23.04,0:10:28.07,Default,,0000,0000,0000,,Is FGX is greater than or equal\Nto minus 8?
Dialogue: 0,0:10:29.82,0:10:35.14,Default,,0000,0000,0000,,Now FG of X is range isn't the\Nsame as F, of X is range.
Dialogue: 0,0:10:36.03,0:10:38.78,Default,,0000,0000,0000,,But it certainly contained in\Nthe range of F of X.
Dialogue: 0,0:10:39.81,0:10:43.17,Default,,0000,0000,0000,,So these two examples show you\Nthat if you're given a composed
Dialogue: 0,0:10:43.17,0:10:46.25,Default,,0000,0000,0000,,function, the range must lie\Nwithin the range of the second
Dialogue: 0,0:10:46.25,0:10:49.33,Default,,0000,0000,0000,,function you apply, but doesn't\Nhave to be all of it.