> ## Content Index
> Fetch the complete content index at: https://www.esheets.io/llms.txt
> Use this file to discover other available public pages before exploring further.

# Composite Functions
- URL: https://www.esheets.io/composite-functions/
- Published: 2026-10-17T18:59:03.000Z
- Updated: 2026-10-17T19:08:09.000Z
- Author: Richard Linnington
- Tags: Algebra, Maths

Practise composite functions with eight self-marking GCSE Higher questions. Work from right to left to evaluate and form fg(x), gf(x), repeated functions and three-function compositions, then solve an equation involving two different compositions. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This eight-question GCSE Higher worksheet builds from numerical compositions to algebraic compositions and equations.

You will use two and three functions, including linear and quadratic rules, and compare fg(x) with gf(x).

#### What you’ll practise

- evaluating fg(a), gf(a) and a repeated function;
- forming and simplifying composite functions;
- working through three functions from right to left;
- solving fg(x) = gf(x).

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Work from right to left: fg(x) means f(g(x)), not f(x) × g(x). Every numbered question is worth one mark.

## Topic guide

### What is a composite function?

A composite function uses the output of one function as the input of another. The notation fg(x) means f(g(x)): apply g first, then apply f. It does **not** mean f(x) multiplied by g(x).

### A numerical example

Let f(x) = 2x + 3 and g(x) = x − 4\. To find fg(5), first calculate g(5) = 1\. Then use that output in f: f(1) = 5\. Therefore fg(5) = 5.

### Why order matters

Using the same functions, gf(5) means g(f(5)). First f(5) = 13, then g(13) = 9, so gf(5) = 9\. In this example fg(5) and gf(5) are different. Always read a composition from right to left.

### Composing a quadratic and a linear function

If f(x) = 3x − 1 and g(x) = x² + 2, then fg(x) = f(x² + 2) = 3(x² + 2) − 1 = 3x² + 5.

In the other order, gf(x) = g(3x − 1) = (3x − 1)² + 2\. The whole linear expression must be squared, so brackets are important.

### Three functions

For fgh(x), begin with h(x), put that result into g, then put the new result into f. Keep each substituted expression in brackets until you simplify.

### Solving fg(x) = gf(x)

Let f(x) = 2x + 2 and g(x) = x² + 2\. Then fg(x) = 2x² + 6, while gf(x) = (2x + 2)² + 2 = 4x² + 8x + 6\. Equating them gives 2x² + 8x = 0\. Factorising gives 2x(x + 4) = 0, hence x = 0 or x = −4.

### Useful checks

- Start with the function nearest to x or the number.
- Use brackets when substituting one expression into another.
- Do not assume fg(x) and gf(x) are equal.
- Expand squared brackets carefully, including the middle term.