The composition of two functions G&F is the function we get from
doing at 1st and then G. So if we let F of X.
Be given by F of X equals X squared.
And we let G of X.
BX +3.
Then GF of X is given by. Well, we do F of X first and
that's X squared.
Then we apply G to the whole of that.
Now, since G just adds 3 onto a
number. G of X squared is X squared +3.
So GF of X is X squared +3.
Here's another example.
This time will let F of X.
Be given by F of X equals 2X.
And we'll have G of X.
Being East The X.
Now to get GFX again, we write down what F of X is.
That was 2X.
And then we apply G to the whole of that.
Now G of X is E to the X.
So G of two X is E to the 2X.
So again we have here GF of X is E to the 2X.
Sometimes the composition of two functions is called the function
of a function, and here's another way of writing it down.
We sometimes writes GF of X as G Little Circle F
of X.
Now you don't have to use this notation yourself, but you
might see it in classes or workbooks, so it's important
to know what it means.
The order in which we compose functions can make a big
difference. If you remember our first example, we had F of X
being X squared.
And G of X.
Being X +3.
We'll see what happens when we do G 1st and then F, so this is
FG of X.
So this time we write down what G of X is first, which is X +3.
Then we apply F to the whole of that.
Now applying the F to something just makes it square, so F of X
+3 is X plus three all squared.
And that X squared plus 6X plus 9.
And if you remember, GF of X was just X squared +3.
So you can see here that changing the order in which
we compose these functions makes a big difference.
Now we've looked at how to compose functions. We look at
how to decompose functions.
If I have a function like each of X equals E to the 2X.
We've already seen that we can write H as a composition
of two functions.
H of X was equal to GF of X, where F of X was
2X and G of X was E to the X.
Let's decompose another function. This time will have
function each of X.
Equals the cube roots.
Of 2X plus one.
The first here we send X to 2X Plus One and then we cube root
the whole thing. So if we let F of XB2X plus one then H of X.
Is the cube root?
Oh, this is X.
So then if we sat G of X.
To be cube root X.
We get that HX.
Is G.
Of F of X. So that's GF of X.
Being able to decompose functions will be very
important when you come to do things like integration and
Differentiation.
Now there is some pairs of functions that can't be
composed. Let's have F of X being minus X squared.
And we'll have G of X.
Being natural log of X.
Now F of X.
Is less than or equal to 0?
Whichever X you pick.
But G of X is only defined if X is greater than 0.
So G of F of X.
Doesn't exist.
There are also some pairs of functions that can be
composed with some ex, but not for other ex.
This time will have F of X equals 4X minus 6.
And we'll have geovax.
Bing square root of X.
Now G of X is only defined if X is greater than or equal to 0.
So GF of X can only be defined if F of X is greater
than or equal to 0.
And F of X is greater than or equal to 0.
Only if X is greater than or equal to three over 2.
So GF of X is only defined.
4X greater than or equal to three over 2.
The values of X for which GF is defined is called the
domain of GF.
The range of a function is the set of all the values of
function can take. So for example, if F
of X was each the X.
The range.
Is given by F of X is greater than zero because E to the X can
only be greater than 0.
Similarly, if we have F of X being say, Sign X.
Then the range.
Is FX.
Is between plus one and minus one.
The range of a composite function must lie within the
range of the second function we
apply. Here's an example to show
you this. Will put down F of X equals X
minus 8 and G of X?
Equals X squared.
Now for F of X.
The range.
Is well F of X can take any value.
But for G of X.
Ranges G of X is greater than or equal to 0.
Now let's look at the range of the composed function GF.
GFX is well F of X is X minus 8.
And G of X squares it, so GF of X is X minus 8
squared. Which is X squared minus 16X plus
64.
But since GF of X is something squared.
And all square numbers are greater than or equal to 0.
GF of X is greater than or equal to 0.
So the range here.
Is GF of X is greater than or equal to 0?
So you can see here that the range of GF is the range of G.
Now let's look at what happens when he composed
and the other way around.
So this time FG books.
What again G of X is X squared?
And applying eh? The whole of that gives us X squared minus 8.
Now, since G of X is greater than or equal to 0.
And here we're taking away 8.
This whole thing must be greater than or equal to minus 8.
So the range.
Is FGX is greater than or equal to minus 8?
Now FG of X is range isn't the same as F, of X is range.
But it certainly contained in the range of F of X.
So these two examples show you that if you're given a composed
function, the range must lie within the range of the second
function you apply, but doesn't have to be all of it.