Functions, Domain and Range — Pure Mathematics 1 (9709) — Mathematics

Introduction

Functions are one of the most fundamental building blocks of A-Level Pure Mathematics. Every major topic in 9709 — from curve sketching to integration — relies on a clear understanding of what a function is and how it behaves. This note addresses the core vocabulary and skills set out in syllabus objective 1.2: function, domain, range, one-one function, inverse function, and composition of functions. Exam questions routinely test whether you can state a domain or range precisely, determine whether an inverse exists, find it algebraically, and evaluate composite functions. Mastering these ideas early pays dividends throughout the entire course.

Core Concept

What is a Function?

A function $f$ is a rule that maps each element of a set called the domain to exactly one element of another set. The set of all output values actually produced is called the range (or image set).

Every input has one and only one output.
Multiple inputs can share the same output (e.g. $f(x)=x^2$ maps both $2$ and $-2$ to $4$ ).
If any input maps to more than one output, the rule is not a function.

The vertical line test is a graphical check: if any vertical line crosses the graph more than once, the rule is not a function.

Domain and Range

The domain is the set of permitted input values. When a domain is not explicitly stated, it is taken to be the largest subset of $\mathbb{R}$ for which the rule is defined (the natural domain).

The range is the set of all output values $f(x)$ produces as $x$ varies over the domain.

One-One Functions

A function is one-one (injective) if every element of the range comes from exactly one element of the domain — no two different inputs give the same output.

f(x_1) = f(x_2) \implies x_1 = x_2

Graphically, a one-one function passes the horizontal line test: no horizontal line crosses the graph more than once.

Why it matters: A function has an inverse function if and only if it is one-one. Restricting the domain of a many-one function can make it one-one.

Inverse Function

For a one-one function $f$ with domain $A$ and range $B$ , the inverse function $f^{-1}$ maps each element of $B$ back to the unique element of $A$ from which it came.

f^{-1}: B \to A, \qquad f^{-1}(f(x)) = x \text{ for all } x \in A

Key facts:

The domain of $f^{-1}$ equals the range of $f$ .
The range of $f^{-1}$ equals the domain of $f$ .
The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$ .

To find $f^{-1}(x)$ algebraically:

Write $y = f(x)$ .
Rearrange to make $x$ the subject.
Replace $x$ with $f^{-1}(x)$ and $y$ with $x$ .
State the domain of $f^{-1}$ .

Composition of Functions

The composite function $gf$ (also written $g \circ f$ ) means "apply $f$ first, then $g$ ":

gf(x) = g(f(x))

Order matters: $gf \neq fg$ in general.

For $gf$ to be defined, the range of $f$ must be a subset of the domain of $g$ .

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