LearnYMA01WMA13 · Pure Mathematics 3Algebra and functionsOne-one and inverse functions

Algebra and functions · Subtopic 5 of 7

One-one and Inverse Functions

Which functions can be undone, how to find $f^{-1}(x)$ , why the domain of the inverse is the range of the function, and how to invert a quadratic on a restricted domain.

Cambridge 9709 Paper 1Edexcel YMA01 WMA13Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2≈ 45 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain what a one-one function is, and use the horizontal line test.
Restrict the domain of a quadratic so that it becomes one-one.
Find $f^{-1}(x)$ by swapping $x$ and $y$ and rearranging.
State the domain and range of $f^{-1}$ from the range and domain of $f$ .
Choose the correct sign of a square root when inverting a quadratic.

Pause and think

Warm-up: the rule "double, then add $3$ " turns $5$ into $13$ . Which number turns into $21$ ? How did you work it out?

Section 2 of 14

Core idea

Start hereUndoing a code

A spy encodes a number by "double it, then add 3": $7$ becomes $17$ . Her partner receives $17$ and must decode it: "subtract 3, then halve" gives $7$ back. The decoder is the inverse function. But some codes can't be undone. "Square it" sends both $3$ and $-3$ to $9$ : given $9$ , which was it? Inverses only exist for rules that never send two inputs to the same output.

The big idea

The inverse $f^{-1}$ undoes $f$ : $f^{-1}(f(x))=x$ . It exists only if $f$ is one-one. To find it, write $y=f(x)$ , rearrange to make $x$ the subject, then swap the letters. The domain of $f^{-1}$ is the range of $f$ , and vice versa.

1. Undoing step by step

f and its inverse undo each other

The inverse runs the steps backwards and in reverse order.

To undo a sequence of operations, undo each one in reverse order (like taking off socks and shoes: shoes come off first).

2. When an inverse exists

y = x² is many-one; restricting it to x ≥ 0 makes it one-one

With the whole curve,

y=4

comes from

x=2

and

x=-2

. Keep only

x\ge0

and every output has just one input, so an inverse

f^{-1}(x)=\sqrt x

exists.

$f$ must be one-one: every horizontal line meets the graph at most once.
A many-one function (like $x^2$ on $\mathbb R$ ) can be made one-one by restricting the domain, e.g. to $x\ge0$ .

3. Finding an inverse algebraically

Worked example $f(x)=\dfrac{3x-1}{2}$

Find $f^{-1}(x)$ for $f(x)=\dfrac{3x-1}2$ , $x\in\mathbb R$ .

Goal: A formula for $f^{-1}(x)$ .
Strategy: Let $y=f(x)$ , make $x$ the subject, swap $x$ and $y$ .

$y=\frac{3x-1}2$
Name the output $y$ .
$2y=3x-1$
Multiply both sides by $2$ .
$2y+1=3x$
Add $1$ .
$x=\frac{2y+1}3$
Divide by $3$ .
$f^{-1}(x)=\frac{2x+1}3$
Rename: inputs are called $x$ .

Answer

f^{-1}(x)=\dfrac{2x+1}{3}

✓ Check

f(5)=7

;

f^{-1}(7)=\frac{15}3=5

✓.

Worked exampleA quadratic with restricted domain

$f(x)=(x-2)^2+1$ for $x\ge2$ . Find $f^{-1}(x)$ and state its domain.

Goal: Formula and domain.
Strategy: Rearrange; choose the square-root sign using the restricted domain.

$y=(x-2)^2+1$
Start.
$y-1=(x-2)^2$
Subtract $1$ .
$x-2=+\sqrt{y-1}$
Square root. Since $x\ge2$ , $x-2\ge0$ : take the positive root only.
$f^{-1}(x)=2+\sqrt{x-1}$
Add $2$ ; swap letters.
Range of $f$ : $f(x)\ge1$ (vertex $(2,1)$ ), so domain of $f^{-1}$ is $x\ge1$ .

Answer

f^{-1}(x)=2+\sqrt{x-1}

x\ge1

✓ Check

f(5)=10

;

f^{-1}(10)=2+3=5

✓.

Common misconception $f^{-1}(x)$ is not $\dfrac1{f(x)}$ The

-1

means "inverse", not a power. For

f(x)=2x

f^{-1}(x)=\frac x2

, but

\frac1{f(x)}=\frac1{2x}

Finding an inverse

Drawing diagram…

Always finish by stating the domain of the inverse.

Pause and think

Why can't $f(x)=x^2$ , $x\in\mathbb R$ have an inverse?

Summary

Only one-one functions have inverses.
$y=f(x)$ , make $x$ the subject, swap.
Domain of $f^{-1}$ = range of $f$ .

Inverse function: $f^{-1}$ , which undoes $f$ .
One-one: Each output comes from exactly one input.

Horizontal line test

f\ \text{is one-one}\iff\text{every horizontal line meets }y=f(x)\text{ at most once}

A quadratic is never one-one on all real numbers, but it becomes one-one if you keep only one side of its vertex. For $f(x)=(x-2)^2-3$ , the domain $x\ge2$ (or $x\le2$ ) works.

HistoryWho wrote f⁻¹ first? The notation

f^{-1}

was introduced in 1813 by the English astronomer John Herschel. He also wrote

\sin^{-1}x

for the inverse sine, and two centuries of students have since confused it with

\dfrac1{\sin x}

Fun factSecret codes need inverses. A code that shifts every letter three places (

A\to D

B\to E

, …) is a function on letters, and decoding is its inverse: shift back three. If two letters were ever coded the same way, the message could not be decoded. That is why every encryption method must be one-one.

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