Functions (Edexcel IGCSE Maths): Revision notes

Section 1

What is function notation?

A function is a rule that takes an input and produces exactly one output. We write $f(x)$ to mean "the output of function $f$ when the input is $x$ ".

For example, if $f(x) = 2x + 3$ , then:

$f(4) = 2(4) + 3 = 11$
$f(-1) = 2(-1) + 3 = 1$
$f(a) = 2a + 3$ (you can substitute an expression, not just a number)

To evaluate $f(x)$ at a given input, replace every $x$ in the rule with that input and simplify.

Key termsfunctioninputoutput

Common mistake

Do not confuse $f(x)$ with $f \times x$ — the bracket means "input", not multiplication.

Example

If $g(x) = x^2 - 5$ , find $g(3)$ : substitute $x=3$ , giving $g(3) = 9 - 5 = 4$ .

Section 2

How do I find the domain and range?

The domain is the set of all valid inputs (x-values) a function can accept. The range is the set of all possible outputs (y-values) the function can produce.

For most GCSE functions (linear, quadratic), the domain is "all real numbers" unless restricted by the question or by the function type. Watch for restrictions:

Fractions: the denominator cannot equal zero, e.g. $f(x) = \dfrac{1}{x-2}$ has domain $x \neq 2$ .
Square roots: the expression inside must be $\geq 0$ , e.g. $f(x) = \sqrt{x-3}$ has domain $x \geq 3$ .

For the range, consider the shape of the graph. A quadratic $f(x) = x^2$ has range $f(x) \geq 0$ because squaring never gives a negative result. If a domain is restricted (e.g. $x \geq 1$ ), find the range by considering only that part of the graph.

Key termsdomainrange

Exam tip

Always check for values that make a denominator zero or a square root negative — these are excluded from the domain.

Example

$f(x) = x^2 + 1$ for $x \geq 2$ : since $x \geq 2$ is increasing, the minimum output is $f(2) = 5$ , so the range is $f(x) \geq 5$ .

Section 3

How do composite functions work?

A composite function combines two functions so the output of one becomes the input of the other. $fg(x)$ means "do $g$ first, then feed the result into $f$ ": $fg(x) = f(g(x))$ .

Order matters — $fg(x)$ is usually different from $gf(x)$ .

Example: if $f(x) = x + 1$ and $g(x) = 3x$ , then:

$fg(x) = f(g(x)) = f(3x) = 3x + 1$
$gf(x) = g(f(x)) = g(x+1) = 3(x+1) = 3x + 3$

To evaluate a composite function at a number, e.g. $fg(2)$ , work from the inside out: find $g(2)$ first, then substitute that result into $f$ .

Key termscomposite function

Common mistake

$fg(x)$ means apply $g$ first then $f$ — the letter closest to $x$ acts first, the opposite of how it might read.

Think of it like this

Think of it like getting dressed: $g$ puts on your socks, $f$ puts on your shoes. You must do $g$ (socks) before $f$ (shoes) — the order is fixed and matters.

Section 4

How do I find an inverse function?

The inverse function $f^{-1}(x)$ reverses what $f(x)$ does — it takes an output back to its original input. Graphically, $f^{-1}(x)$ is the reflection of $f(x)$ in the line $y = x$ .

Method to find $f^{-1}(x)$ :

Write $y = f(x)$ .
Rearrange to make $x$ the subject.
Swap $x$ and $y$ .
Rewrite as $f^{-1}(x) = \ldots$

Example: $f(x) = 2x - 5$

$y = 2x - 5$
$y + 5 = 2x \Rightarrow x = \dfrac{y+5}{2}$
Swap: $y = \dfrac{x+5}{2}$
So $f^{-1}(x) = \dfrac{x+5}{2}$

A useful check: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all valid $x$ .

Key termsinverse function

Exam tip

Always check your inverse by substituting a simple number: if $f(3) = 1$ , confirm $f^{-1}(1) = 3$ .

Common mistake

$f^{-1}(x)$ is NOT the same as $\dfrac{1}{f(x)}$ — the $-1$ here means inverse function, not reciprocal.

Must Know

$f(x)$ means substitute $x$ into the rule for $f$ ; brackets show the input, not multiplication.
Domain = valid inputs; exclude values making a denominator zero or a square root negative.
Range = possible outputs; consider the shape/turning point of the graph, especially with restricted domains.
$fg(x) = f(g(x))$ : apply the function closest to $x$ first.
To find $f^{-1}(x)$ : write $y=f(x)$ , rearrange for $x$ , swap $x$ and $y$ .
Check inverses using $f(f^{-1}(x)) = x$ .

That's the notes covered.

Carry on to the next subtopic.