0:00:02.200,0:00:06.736
The composition of two functions[br]G&F is the function we get from

0:00:06.736,0:00:11.650
doing at 1st and then G. So if[br]we let F of X.

0:00:13.470,0:00:16.575
Be given by F of X[br]equals X squared.

0:00:17.620,0:00:19.120
And we let G of X.

0:00:20.780,0:00:22.670
BX +3.

0:00:24.320,0:00:30.365
Then GF of X is given by.[br]Well, we do F of X first and

0:00:30.365,0:00:31.574
that's X squared.

0:00:32.940,0:00:34.965
Then we apply G to the[br]whole of that.

0:00:38.230,0:00:40.678
Now, since G just adds 3 onto a

0:00:40.678,0:00:45.194
number. G of X squared is[br]X squared +3.

0:00:46.430,0:00:49.174
So GF of X is X squared +3.

0:00:50.890,0:00:52.090
Here's another example.

0:00:54.610,0:00:56.444
This time will let F of X.

0:00:57.050,0:00:59.914
Be given by F of X equals 2X.

0:01:01.310,0:01:02.660
And we'll have G of X.

0:01:04.120,0:01:05.528
Being East The X.

0:01:07.710,0:01:12.936
Now to get GFX again, we write[br]down what F of X is.

0:01:13.870,0:01:15.070
That was 2X.

0:01:17.100,0:01:19.310
And then we apply G[br]to the whole of that.

0:01:22.300,0:01:24.595
Now G of X is E to the X.

0:01:25.160,0:01:28.500
So G of two X is E to the 2X.

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So again we have here GF of X is[br]E to the 2X.

0:01:35.060,0:01:38.120
Sometimes the composition of two[br]functions is called the function

0:01:38.120,0:01:41.486
of a function, and here's[br]another way of writing it down.

0:01:42.770,0:01:49.634
We sometimes writes GF of[br]X as G Little Circle F

0:01:49.634,0:01:50.882
of X.

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Now you don't have to use this[br]notation yourself, but you

0:01:55.764,0:01:58.504
might see it in classes or[br]workbooks, so it's important

0:01:58.504,0:01:59.874
to know what it means.

0:02:03.920,0:02:06.890
The order in which we compose[br]functions can make a big

0:02:06.890,0:02:12.392
difference. If you remember our[br]first example, we had F of X

0:02:12.392,0:02:13.628
being X squared.

0:02:15.080,0:02:16.148
And G of X.

0:02:17.530,0:02:18.919
Being X +3.

0:02:21.040,0:02:26.890
We'll see what happens when we[br]do G 1st and then F, so this is

0:02:26.890,0:02:28.060
FG of X.

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So this time we write down what[br]G of X is first, which is X +3.

0:02:36.020,0:02:38.864
Then we apply F to[br]the whole of that.

0:02:40.410,0:02:45.772
Now applying the F to something[br]just makes it square, so F of X

0:02:45.772,0:02:48.453
+3 is X plus three all squared.

0:02:49.860,0:02:54.068
And that X squared plus[br]6X plus 9.

0:02:55.450,0:03:00.550
And if you remember, GF of X was[br]just X squared +3.

0:03:02.440,0:03:05.135
So you can see here that[br]changing the order in which

0:03:05.135,0:03:07.095
we compose these functions[br]makes a big difference.

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Now we've looked at how to[br]compose functions. We look at

0:03:18.056,0:03:19.240
how to decompose functions.

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If I have a function like each[br]of X equals E to the 2X.

0:03:27.360,0:03:30.649
We've already seen that we[br]can write H as a composition

0:03:30.649,0:03:31.546
of two functions.

0:03:33.130,0:03:40.984
H of X was equal to GF[br]of X, where F of X was

0:03:40.984,0:03:46.594
2X and G of X was E[br]to the X.

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Let's decompose another[br]function. This time will have

0:03:57.346,0:03:59.214
function each of X.

0:04:00.480,0:04:01.860
Equals the cube roots.

0:04:02.890,0:04:04.938
Of 2X plus one.

0:04:07.330,0:04:14.830
The first here we send X to 2X[br]Plus One and then we cube root

0:04:14.830,0:04:22.830
the whole thing. So if we let F[br]of XB2X plus one then H of X.

0:04:24.380,0:04:25.400
Is the cube root?

0:04:26.440,0:04:27.508
Oh, this is X.

0:04:29.360,0:04:31.488
So then if we sat G of X.

0:04:32.660,0:04:34.150
To be cube root X.

0:04:35.770,0:04:37.498
We get that HX.

0:04:38.910,0:04:40.470
Is G.

0:04:42.000,0:04:47.229
Of F of X. So that's[br]GF of X.

0:04:48.420,0:04:50.988
Being able to decompose[br]functions will be very

0:04:50.988,0:04:54.198
important when you come to do[br]things like integration and

0:04:54.198,0:04:54.519
Differentiation.

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Now there is some pairs of[br]functions that can't be

0:05:05.810,0:05:13.149
composed. Let's have F of[br]X being minus X squared.

0:05:14.000,0:05:15.548
And we'll have G of X.

0:05:16.930,0:05:20.500
Being natural log of X.

0:05:22.650,0:05:24.358
Now F of X.

0:05:25.660,0:05:27.550
Is less than or equal to 0?

0:05:28.270,0:05:29.938
Whichever X you pick.

0:05:31.190,0:05:36.299
But G of X is only defined if X[br]is greater than 0.

0:05:37.410,0:05:40.356
So G of F of X.

0:05:41.550,0:05:42.820
Doesn't exist.

0:05:53.750,0:05:56.320
There are also some pairs of[br]functions that can be

0:05:56.320,0:05:58.633
composed with some ex, but[br]not for other ex.

0:05:59.720,0:06:06.463
This time will have F of[br]X equals 4X minus 6.

0:06:07.450,0:06:08.810
And we'll have geovax.

0:06:11.160,0:06:13.650
Bing square root of X.

0:06:15.290,0:06:20.538
Now G of X is only defined if X[br]is greater than or equal to 0.

0:06:21.510,0:06:25.570
So GF of X can only be[br]defined if F of X is greater

0:06:25.570,0:06:27.020
than or equal to 0.

0:06:28.660,0:06:33.632
And F of X is greater than or[br]equal to 0.

0:06:34.350,0:06:38.394
Only if X is greater than or[br]equal to three over 2.

0:06:42.140,0:06:46.606
So GF of X is only[br]defined.

0:06:50.320,0:06:54.739
4X greater than or equal to[br]three over 2.

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The values of X for which[br]GF is defined is called the

0:07:01.364,0:07:02.330
domain of GF.

0:07:08.950,0:07:12.343
The range of a function is the[br]set of all the values of

0:07:12.343,0:07:16.660
function can take.[br]So for example, if F

0:07:16.660,0:07:19.972
of X was each the X.

0:07:21.630,0:07:22.590
The range.

0:07:24.330,0:07:29.818
Is given by F of X is greater[br]than zero because E to the X can

0:07:29.818,0:07:31.533
only be greater than 0.

0:07:32.590,0:07:37.551
Similarly, if we have F of X[br]being say, Sign X.

0:07:40.720,0:07:41.740
Then the range.

0:07:45.800,0:07:47.420
Is FX.

0:07:48.880,0:07:52.926
Is between plus one and[br]minus one.

0:08:00.460,0:08:03.310
The range of a composite[br]function must lie within the

0:08:03.310,0:08:05.020
range of the second function we

0:08:05.020,0:08:07.040
apply. Here's an example to show

0:08:07.040,0:08:12.528
you this. Will put[br]down F of X equals X

0:08:12.528,0:08:15.624
minus 8 and G of X?

0:08:16.990,0:08:18.268
Equals X squared.

0:08:19.920,0:08:21.260
Now for F of X.

0:08:21.860,0:08:22.780
The range.

0:08:25.730,0:08:27.935
Is well F of X can take[br]any value.

0:08:35.350,0:08:37.180
But for G of X.

0:08:38.470,0:08:42.837
Ranges G of X is greater than or[br]equal to 0.

0:08:44.420,0:08:47.511
Now let's look at the range of[br]the composed function GF.

0:08:49.190,0:08:55.140
GFX is well F of X[br]is X minus 8.

0:08:57.670,0:09:05.006
And G of X squares it, so[br]GF of X is X minus 8

0:09:05.006,0:09:12.255
squared. Which is X[br]squared minus 16X plus

0:09:12.255,0:09:13.120
64.

0:09:15.360,0:09:19.480
But since GF of X is[br]something squared.

0:09:20.620,0:09:23.205
And all square numbers are[br]greater than or equal to 0.

0:09:24.050,0:09:27.180
GF of X is greater[br]than or equal to 0.

0:09:28.450,0:09:29.590
So the range here.

0:09:32.320,0:09:36.687
Is GF of X is greater than or[br]equal to 0?

0:09:38.260,0:09:43.150
So you can see here that the[br]range of GF is the range of G.

0:09:44.510,0:09:46.373
Now let's look at what[br]happens when he composed

0:09:46.373,0:09:47.408
and the other way around.

0:09:48.780,0:09:50.980
So this time FG books.

0:09:53.990,0:09:56.478
What again G of X is X squared?

0:09:59.150,0:10:04.880
And applying eh? The whole of[br]that gives us X squared minus 8.

0:10:06.410,0:10:09.410
Now, since G of X is greater[br]than or equal to 0.

0:10:10.100,0:10:11.786
And here we're taking away 8.

0:10:12.590,0:10:16.790
This whole thing must be greater[br]than or equal to minus 8.

0:10:18.820,0:10:19.948
So the range.

0:10:23.040,0:10:28.070
Is FGX is greater than or equal[br]to minus 8?

0:10:29.820,0:10:35.145
Now FG of X is range isn't the[br]same as F, of X is range.

0:10:36.030,0:10:38.780
But it certainly contained in[br]the range of F of X.

0:10:39.810,0:10:43.170
So these two examples show you[br]that if you're given a composed

0:10:43.170,0:10:46.250
function, the range must lie[br]within the range of the second

0:10:46.250,0:10:49.330
function you apply, but doesn't[br]have to be all of it.