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Algebra · Subtopic 7 of 11

Functions

A function $f(x)$ is a rule turning an input into an output. A composite $fg(x)$ means "do $g$ , then $f$ ". The inverse $f^{-1}$ undoes $f$ : swap $x$ and $y$ and rearrange. Some inputs can't be used, such as values that make you divide by zero.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use function notation to evaluate and solve.
Form and evaluate composite functions such as $fg(x)$ and $ff(x)$ .
Find an inverse function $f^{-1}(x)$ .
Identify values that must be excluded from the domain.

Pause and think

A machine doubles a number, then adds $3$ . It outputs $17$ . What went in? How did you work it out?

Section 2 of 14

Core idea

Start hereA machine

Think of a function as a machine: drop in a number, it does something fixed (double it, add 3), and out comes the answer. Run it backwards and you get the inverse; feed one machine into another and you get a composite.

The big idea

$f(3)$ : put in 3. $f^{-1}$ reverses $f$ . $fg(x)=f(g(x))$ : $g$ first, then $f$ .

1. Evaluating and inverses

A function machine and its inverse

To undo

f

, reverse the steps and the order:

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

Worked example $f(x)=2x+3$

$f(4)$ and $f^{-1}(x)$ .

Goal: Value; inverse.
Strategy: Swap the roles: solve $y=2x+3$ for $x$ .

$f(4)=11$
Substitute.
$y=2x+3\Rightarrow x=\frac{y-3}2$
Undo: subtract 3, halve.
$f^{-1}(x)=\frac{x-3}2$
Rename the input $x$ .

Answer

11

;

\frac{x-3}2

✓ Check

f^{-1}(11)=4

✓.

2. Composite functions

Worked example $f(x)=2x+3$ , $g(x)=x^2$

$fg(x)$ , $gf(x)$ , $gf(1)$ .

Goal: Composites.
Strategy: Inner function first.

$fg(x)=f(x^2)=2x^2+3$
Square first.
$gf(x)=g(2x+3)=(2x+3)^2$
Double-add first.
$gf(1)=5^2=25$
Evaluate.

Answer

2x^2+3

;

(2x+3)^2

;

25

✓ Check

fg(1)=5\neq25

: order matters ✓.

Common misconception $f^{-1}(x)\neq\frac1{f(x)}$ The inverse undoes

f

; it isn't the reciprocal.

Pause and think

$f(x)=\frac x4-1$ . Find $f^{-1}(x)$ .

Summary

Inverse: rearrange.
$fg$ : $g$ first.

Inverse function: Reverses a function.
Composite function: Function of a function.

HistoryWho wrote f(x) first? The notation

f(x)

was introduced by Leonhard Euler in

1734

. Before that, mathematicians described functions in words. Euler also gave us

e

i

and

\Sigma

, and popularised

\pi

: perhaps the greatest inventor of notation in history.

Fun factEncryption is a function. A secret code is a function from plain messages to coded ones, and decoding is its inverse. Modern encryption uses functions that are easy to compute but practically impossible to invert without a secret key.

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