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The composition of two functions
G&F is the function we get from

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doing at 1st and then G. So if
we let F of X.

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Be given by F of X
equals X squared.

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And we let G of X.

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BX +3.

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Then GF of X is given by.
Well, we do F of X first and

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that's X squared.

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Then we apply G to the
whole of that.

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Now, since G just adds 3 onto a

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number. G of X squared is
X squared +3.

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So GF of X is X squared +3.

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Here's another example.

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This time will let F of X.

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Be given by F of X equals 2X.

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And we'll have G of X.

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Being East The X.

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Now to get GFX again, we write
down what F of X is.

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That was 2X.

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And then we apply G
to the whole of that.

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Now G of X is E to the X.

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So G of two X is E to the 2X.

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

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Sometimes the composition of two
functions is called the function

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of a function, and here's
another way of writing it down.

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We sometimes writes GF of
X as G Little Circle F

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of X.

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

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might see it in classes or
workbooks, so it's important

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to know what it means.

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The order in which we compose
functions can make a big

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difference. If you remember our
first example, we had F of X

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being X squared.

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And G of X.

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Being X +3.

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We'll see what happens when we
do G 1st and then F, so this is

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FG of X.

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

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Then we apply F to
the whole of that.

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Now applying the F to something
just makes it square, so F of X

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+3 is X plus three all squared.

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And that X squared plus
6X plus 9.

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And if you remember, GF of X was
just X squared +3.

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So you can see here that
changing the order in which

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we compose these functions
makes a big difference.

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

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how to decompose functions.

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

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We've already seen that we
can write H as a composition

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of two functions.

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H of X was equal to GF
of X, where F of X was

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2X and G of X was E
to the X.

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Let's decompose another
function. This time will have

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function each of X.

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Equals the cube roots.

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Of 2X plus one.

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The first here we send X to 2X
Plus One and then we cube root

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the whole thing. So if we let F
of XB2X plus one then H of X.

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Is the cube root?

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Oh, this is X.

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So then if we sat G of X.

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To be cube root X.

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We get that HX.

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Is G.

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Of F of X. So that's
GF of X.

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Being able to decompose
functions will be very

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important when you come to do
things like integration and

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Differentiation.

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

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composed. Let's have F of
X being minus X squared.

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And we'll have G of X.

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Being natural log of X.

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Now F of X.

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Is less than or equal to 0?

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Whichever X you pick.

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But G of X is only defined if X
is greater than 0.

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So G of F of X.

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Doesn't exist.

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There are also some pairs of
functions that can be

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composed with some ex, but
not for other ex.

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This time will have F of
X equals 4X minus 6.

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And we'll have geovax.

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Bing square root of X.

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Now G of X is only defined if X
is greater than or equal to 0.

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So GF of X can only be
defined if F of X is greater

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than or equal to 0.

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And F of X is greater than or
equal to 0.

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Only if X is greater than or
equal to three over 2.

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So GF of X is only
defined.

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4X greater than or equal to
three over 2.

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The values of X for which
GF is defined is called the

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domain of GF.

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The range of a function is the
set of all the values of

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function can take.
So for example, if F

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of X was each the X.

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The range.

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Is given by F of X is greater
than zero because E to the X can

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only be greater than 0.

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Similarly, if we have F of X
being say, Sign X.

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Then the range.

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Is FX.

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Is between plus one and
minus one.

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The range of a composite
function must lie within the

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range of the second function we

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apply. Here's an example to show

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you this. Will put
down F of X equals X

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minus 8 and G of X?

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Equals X squared.

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Now for F of X.

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The range.

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Is well F of X can take
any value.

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But for G of X.

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Ranges G of X is greater than or
equal to 0.

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Now let's look at the range of
the composed function GF.

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GFX is well F of X
is X minus 8.

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And G of X squares it, so
GF of X is X minus 8

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squared. Which is X
squared minus 16X plus

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64.

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But since GF of X is
something squared.

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And all square numbers are
greater than or equal to 0.

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GF of X is greater
than or equal to 0.

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So the range here.

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Is GF of X is greater than or
equal to 0?

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So you can see here that the
range of GF is the range of G.

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Now let's look at what
happens when he composed

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and the other way around.

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So this time FG books.

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What again G of X is X squared?

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And applying eh? The whole of
that gives us X squared minus 8.

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Now, since G of X is greater
than or equal to 0.

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And here we're taking away 8.

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This whole thing must be greater
than or equal to minus 8.

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So the range.

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Is FGX is greater than or equal
to minus 8?

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Now FG of X is range isn't the
same as F, of X is range.

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But it certainly contained in
the range of F of X.

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So these two examples show you
that if you're given a composed

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function, the range must lie
within the range of the second

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function you apply, but doesn't
have to be all of it.