Composite Functions — Pure Mathematics 1 (9709) — Mathematics

Introduction

Composite functions arise whenever one function is applied after another — a situation that appears throughout Pure Mathematics and beyond. In 9709 Paper 1, questions on composite functions regularly ask you to form $gf(x)$ , state its domain and range, or determine whether a composition is valid at all. Mastery of this topic requires precise understanding of domain and range, and a careful eye for the condition that governs when a composition is permitted.

Core Concept

What is a Composite Function?

Given two functions $f$ and $g$ , the composite function $gf$ (read "g of f") means: apply $f$ first, then apply $g$ to the result.

gf(x) = g\bigl(f(x)\bigr)

The output of $f$ becomes the input of $g$ . This is the key insight that drives everything else.

Order matters. In general, $gf \neq fg$ . Always apply the function nearest to $x$ first.

The Existence Condition

The composite $gf$ can only be formed when:

\text{Range of } f \subseteq \text{Domain of } g

That is, every value that $f$ can output must be a valid input for $g$ . If even one value in the range of $f$ falls outside the domain of $g$ , the composition $gf$ is not defined.

Domain and Range of a Composite Function

Once $gf$ is valid:

The domain of $gf$ is the domain of $f$ (since $f$ is applied first).
The range of $gf$ is found by tracing what happens to the range of $f$ through $g$ , or by analysing $gf(x)$ directly.

Identifying the Range in Simple Cases

For standard functions:

Function type	Range (typical)
$f(x) = x^2$ with domain $\mathbb{R}$	$f(x) \geq 0$
$f(x) = x^2$ with domain $x \geq 2$	$f(x) \geq 4$
$f(x) = \sqrt{x}$ with domain $x \geq 0$	$f(x) \geq 0$
$f(x) = \frac{1}{x}$ , $x > 0$	$f(x) > 0$
$f(x) = 2x + 1$ , $x \in \mathbb{R}$	$f(x) \in \mathbb{R}$
$f(x) = (x-1)^2 + 3$ , $x \geq 1$	$f(x) \geq 3$

Sketching or completing the square are standard tools for identifying ranges on the 9709 paper.

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