Functions (Cambridge IGCSE Maths): Revision notes

Section 1

What is a function, domain and range?

A function is a rule that maps every input value to exactly one output value, written using function notation, e.g. $f(x) = 3x - 5$ or $h(x) = 2x^2 + 3$ .

The domain is the set of all possible input values ( $x$ -values)
The range is the set of all possible output values ( $f(x)$ -values) produced
To evaluate $f(a)$ , substitute $a$ in place of $x$ in the function rule

For example, if $f(x) = 3x - 5$ , then $f(4) = 3(4) - 5 = 7$ .

Key termsfunctiondomainrange

Exam tip

Treat $f(x)$ notation exactly like substitution — just replace $x$ with whatever value or expression is inside the brackets.

Section 2

How do you use mapping diagrams to represent functions?

A mapping diagram shows two lists of values — the inputs (domain) on one side and the outputs (range) on the other — connected by arrows showing which output each input produces.

Each arrow represents one input being mapped to exactly one output, consistent with the definition of a function. If any input appears to map to two different outputs, the relationship is not a function.

Key termsmapping diagram

Section 3

How do you find an inverse function?

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$ — it takes an output back to the original input.

To find $f^{-1}(x)$ for $f(x) = 3x - 5$ :

Write $y = 3x - 5$
Swap $x$ and $y$ : $x = 3y - 5$
Rearrange to make $y$ the subject: $y = \frac{x+5}{3}$
State the inverse: $f^{-1}(x) = \frac{x+5}{3}$

Key termsinverse function

Common mistake

$f^{-1}(x)$ does NOT mean $\frac{1}{f(x)}$ — the $-1$ here is inverse-function notation, not a power.

Section 4

How do you form composite functions?

A composite function applies one function to the result of another. $gf(x)$ means "do $f$ first, then apply $g$ to the result": $gf(x) = g(f(x))$ .

For example, if $f(x) = 3x - 5$ and $g(x) = x^2$ :

$gf(x) = g(f(x)) = g(3x-5) = (3x-5)^2$

Whereas $fg(x) = f(g(x)) = f(x^2) = 3x^2 - 5$ — note that $gf(x) \neq fg(x)$ in general.

Key termscomposite function

Exam tip

Read $gf(x)$ from right to left: the function closest to $x$ is applied first.

Must Know

A function maps every input to exactly one output; evaluate $f(a)$ by substituting $a$ for $x$
Domain = possible inputs; range = possible outputs
To find $f^{-1}(x)$ : write as $y=$ , swap $x$ and $y$ , then rearrange to make $y$ the subject
$f^{-1}(x)$ is inverse notation, not $\frac{1}{f(x)}$
Composite function $gf(x) = g(f(x))$ applies $f$ first, then $g$ — order matters, so $gf(x) \neq fg(x)$ generally

That's the notes covered.

Carry on to the next subtopic.