A function is a mathematical expression which changes one number into another. It always changes a number the same way.

A composite function is a combination of two functions, where you apply the first function and get an answer, and then fill that answer into the second function.

Let's look at some examples. Here are two simple functions, which we'll label  f and  g:

f(x) = 2x + 5                 g(x) = 3x2


A composite function looks like this:   f º g (x)

The small circle in   f º g (x) means that it is a composite function of  f and  g


f º g (x)  means    f (g(x) ), which tells you to work out  g(x) first, and then fill that answer into  f.


Let's find   f º g (5)

This means find   f (g(5) )

Work out  g(5) first.
Since   g(x) = 3x2, then   g(5) = 3(5)2 = 3(25) = 75

Now find   f(75)
f(x) = 2x + 5 so   f(75) = 2(75) + 5 = 155

So   f º g (5) = 155



Here's another example. Let's use two different functions:

f(x) = 4x2 - 1                 g(x) = 3x + 2


The composite function value we want is  f( g(2) )

First work out  g(2) = 3(2) + 2 = 8
Now work out  f(8) = 4(8)2 - 1 = 4(64) - 1 = 255
So  f( g(2) ) = 255

Notice again that you do the inside function first. Then you fill that answer into the outside function.



It is also possible to work out a simplified version of the composite function, to find a single expression for  f º g (x) that you can use in one step. Here's what we mean: let's use the same two examples we just worked with ...
f(x) = 4x2 - 1                 g(x) = 3x + 2


The composite function value we want is  f( g(2) ), but this time we're going to find the simplified version of  f º g (x) .
We'll do this by filling the expression for g(x) into  f(x), and simplifying the result.

f( g(x) ) = 4(3x + 2)2 - 1   =   4(9x2 + 12x + 4) - 1   =   36x2 + 48x + 16 - 1   =   36x2 + 48x + 15

So  f º g (x)  =  f( g(x) ) = 36x2 + 48x + 15    (This is the simplified composite function)
So  f( g(2) )  =  36(2)2 +48(2) + 15  =  36(4) + 96 + 15  =  255    (This is the answer when x is 2)





Let's do one more to make sure you get the idea. Here are two brand new functions:

f(x) = 4x - 3                 g(x) = 2x2


We want to find the simplified version of the composite function  g º f (x), and then work out  g º f (5).

First, do you notice how this question is different? It's the order of the functions.
If we were doing this question in two steps, we would have to work out  f(5) first, and then fill that answer into  g(x).

In this case, since we want the simplified composite function, we'll be filling function  f into function  g.
We want to first find  g( f(x) ):

g( f(x) ) = 2(4x - 3)2  =  2(16x2 - 24x + 9)  =  32x2 - 48x + 18    (This is the simplified composite function)

g( f(5) )  =  32(5)2 - 48(5) + 18  =  800 - 240 + 18  =   578    (This is the answer when x is 5)







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