# Composite Functions

## Introduction to Composite Functions

Composite functions are functions that are built up of two or more stages. For example:
In general, given that:
The composite function of f and g will be f (g(x)).
This is also often written as .
Although this general rule may seem quite confusing at first, when we go on to do examples, you will realise that it is actually quite simple to solve
composite functions.
## Solving Composite Functions

Let us do a couple of examples in order for you to understand how to deal with composite functions.
As you can see, solving composite functions is relatively easy if you write out all the steps and substitute in the equations. You may also be asked to
substitute numbers into your final equation to get a numerical answer. When you move on in your studies of mathematics, you will also meet situations in
which you have to solve composite functions with more than 2 equations (which are solved in exactly the same way as shown above, just with additional
equations) or find the inverse composite function. However, these two scenarios exceed the scope of this lesson.
## Meeting Inverse Functions

Now that you are pretty confident with functions, it is time to look at inverse functions. Also, make sure to check out our library for more interesting maths
lessons.
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Example 1
Given that:
f(x) = 2x+4 and g(x) = 2x²+1 find the
composite function (f o g)(x) in its simplest
form.
(f o g)(x) = f(g(x))
= f(2x²+1) substitute in g(x)
= 2(2x²+1)+4 make g(x) the x value of f(x)
= 4x² + 2 + 4 expand
= 4x² + 6 simplify

Example 2
Given that:
f(x) = 5x+7 and g(x) = x²+8 find the
composite function (g o f)(x) in its simplest
form.
(g o f)(x) = g(f(x))
= g(5x+7) substitute in g(x)
= (5x+7)² + 8 make g(x) the x value of f(x)
= 25x² + 70x + 49 + 8 expand
= 25x² + 70x + 57 simplify

Chapter 20.1: Learning Outcomes
Students will be introduced to the concept and notation of composite
functions!
Students will learn how to solve composite functions!

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