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# Evaluating a Composite Function

(KS4, Year 10)

## The Lesson

A composite function is a function of a function. It combines two or more functions so that the output of one function becomes the input of another. Evaluating a composite function means putting an input into a composite function, and finding the output it relates to.## Understanding Evaluating a Composite Function

To evaluate a composite function means to see what output an input is mapped to. The image below shows a mapping diagram of a composite function which relates a set of inputs to a set of outputs. If we wanted to evaluate the function when the input is**2**, we would see which output it is mapped to.

## How to Evaluate a Composite Function

Evaluating a composite function is easy.## Question

Two functions are**f(x) = 2x + 1**and

**g(x) = x + 2**. Evaluate the composite function

**fg(x)**at

**x = 2**.

There are two methods for evaluating the composite function.

## Method 1

This method is simpler.## Step-by-Step:

# 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**

**x = 2**, we will pass in**2**as an input, rather than**x**.# 2

Pass the input

This is evaluating the function

**2**into the function**g**.This is evaluating the function

**g(x)**at**x = 2**. Substitute**x = 2**into**g(x)**.
g(x) = x + 2
g(2) = 2 + 2
g(2) = 4

# 3

Pass the

We have found in

**g(2)**into the function**f**.We have found in

**Step 2**that**g(2) = 4**. Evaluate the function**f(x)**at**x = 4**by substituting**x = 4**into**f(x)**.
f(x) = 2x + 1
f(4) = 2 × 4 + 1
f(4) = 8 + 1
f(4) = 9

## Answer:

The composite function**fg(x)**evaluated at

**x = 2**is:

**fg(2) = 9**.

## Method 2

This method is more complicated because you find the composite function before evaluating it. The advantage is that you can then evaluate the composite function at many different values.## Step-by-Step:

# 1

Find the composite function

The composite function

**fg(x)**.fg(x) | Find the left most letter. It is f |

f(x) = 2x + 1 | Write out the function f(x) |

fg(x) = 2g(x) + 1 | Insert a g to the right of the f in the function name and replace x with g(x) |

fg(x) = 2(x + 2) + 1 | Substitute g(x) = x + 2 into the function (put it in brackets) |

fg(x) = 2x + 4 + 1 | Expand the brackets |

fg(x) = 2x + 4 + 1 | Collect the constant terms |

fg(x) = 2x + 5 |

**fg(x) = 2x + 5**.# 2

Evaluate the composite function

**fg(x)**at**x = 2**by substituting**x = 2**into**fg(x)**.
fg(x) = 2x + 5
fg(2) = 2 × 2 + 5
fg(2) = 4 + 5
fg(2) = 9

## Answer:

The composite function**fg(x)**evaluated at

**x = 2**is:

**fg(2) = 9**. Both methods give the same answer: if the functions

**f(x) = 2x + 1**and

**g(x) = x + 2**are combined into a composite function

**fg(x)**and evaluated at

**2**, the answer is

**9**.

## A Note on Notation

If a composite function**fg**is evaluated at a number (e.g.

**2**), this is denoted

**fg(2)**.

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