Inverse Functions — Pure Mathematics 1 (9709)

Core Concept

What is a one-one function?

A function $f$ is one-one (injective) if every element of the range is the image of exactly one element of the domain. In plain terms: no two different inputs produce the same output.

Horizontal Line Test: Draw any horizontal line across the graph of $f$. If every horizontal line meets the graph at most once, the function is one-one.

A function such as $f(x) = x^2$ on $\mathbb{R}$ is not one-one, because $f(2) = f(-2) = 4$ — two inputs share the same output. However, restricting the domain to $x \geq 0$ makes it one-one.

What is an inverse function?

If $f : A \to B$ is one-one (and onto its range), then its inverse $f^{-1} : B \to A$ reverses the mapping:

Graphically, $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.

Domain and range swap:

• Domain of $f^{-1}$ = Range of $f$
• Range of $f^{-1}$ = Domain of $f$

This swap is essential — examiners regularly award a mark specifically for stating the domain of $f^{-1}$.