# How to Find a Composite Function

## Finding a Composite Function

A composite function is a function of a function.

A composite function combines two or more functions so that the output of one function becomes the input of another.

## How to Find a Composite Function

Finding a composite function is easy.

### Question

Find the composite function fg(x), where f(x) = x2 + 1 and g(x) = x + 2.

# 1

Understand the composite function.

In our example, the composite function is fg(x). Reading right to left, this means:

• the input x is passed into the function g

• which is passed into the function f # 2

Find the left most letter.

In our example, the composite function is fg(x). f is the left most letter.

Write out the function f(x)

f(x) = x2 + 1

# 3

We are passing g(x) into this function

• Insert a g to the right of the f in the function name.

fg(x) = x2 + 1

• Replace x with g(x).

fg(x) = g(x)2 + 1

# 4

Substitute g(x) = x + 2 into the function.

fg(x) = (x + 2)2 + 1

Don't forget: Use brackets when substituting an expression. This reminds you that the whole expression is being squared.

# 5

Simplify the equation.

 fg(x) = (x + 2)2 + 1 fg(x) = (x + 2)(x + 2) + 1 (x + 2)2 = (x + 2) × (x + 2) fg(x) = x2 + 4x + 4 + 1 Use the FOIL method to expand the brackets fg(x) = x2 + 4x + 4 + 1 Collect the constant terms fg(x) = x2 + 4x + 5

The composite function fg(x) of f(x) = x2 + 1 and g(x) = x + 2 is:

fg(x) = x2 + 4x + 5.

## Slider

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