Why does a function need a restricted domain to have an inverse?

An inverse must be a function too, so the original mapping has to be one-to-one over the domain you choose.

Pure Maths

A-Level Functions

Functions formalise the idea of an input mapping to exactly one output, and give you the language of domain, range, composition, inverses and transformations.

What you need to know

Domain and range notation
Composite functions fg(x) and the order of operations
Inverse functions and their graphs
Graph transformations: translations, stretches and reflections

Key concepts

fg(x) means apply g first, then f.
The graph of f⁻¹(x) is the reflection of f(x) in the line y = x.
y = f(x + a) shifts left by a; y = f(x) + a shifts up by a.

Common mistakes

Applying composite functions in the wrong order
Forgetting to restrict the domain so an inverse exists
Mixing up inside and outside transformations

Example question

Given $f(x) = 2x + 1$ and $g(x) = x^2$ , find $fg(3)$ .

Show explanation

g(3) = 9, then f(9) = 2 × 9 + 1 = 19.

Practise Functions

4 original questions with worked explanations, mixed across easy, medium and challenging difficulty.

Practise Functions Take the free diagnostic

Functions questions students ask

Why does a function need a restricted domain to have an inverse?: An inverse must be a function too, so the original mapping has to be one-to-one over the domain you choose.