Functions Toolkit | Cambridge O Level Maths Revision Notes 2025

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Introduction to Functions

What is a function?

A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers
- It may be thought of as a mathematical “machine”
- For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"
  - Putting 3 in to the function would give 2 × 3 + 1 = 7
  - Putting -4 in would give 2 × (-4) + 1 = -7
- Putting $x$ in would give $2 x + 1$
The number being put into the function is often called the input
The number coming out of the function is often called the output

What does a function look like?

A function f can be written as f(x) = … or f : x ↦ …
- These two different types of notation mean exactly the same thing
- Other letters can be used. g, h and j are common but any letter can technically be used
  - Normally, a new letter will be used to define a new function in a question
For example, the function with the rule “triple the number and subtract 4” would be written
- $f (x) = 3 x - 4$ or $f : x \mapsto 3 x - 4$
In such cases, $x$ would be the input and $f (x)$ would be the output
Sometimes functions don’t have names like f and are just written as y = …
- eg. $y = 3 x - 4$

How does a function work?

A function has an input $(x)$ and output $(f (x) or y)$
Whatever goes in the bracket (instead of $x$ )with f, replaces the $x$ on the other side
- This is the input
If the input is known, the output can be calculated
- For example, given the function $f (x) = 2 x + 1$
  - $f (3) = 2 \times 3 + 1 = 7$
  - $f (- 4) = 2 \times (- 4) + 1 = - 7$
  - $f (a) = 2 a + 1$

If the output is known, an equation can be formed and solved to find the input
- For example, given the function $f (x) = 2 x + 1$
  - If $f (x) = 15$ , the equation $2 x + 1 = 15$ can be formed
  - Solving this equation gives an input of 7

What is a mapping diagram?

A mapping diagram takes an ‘input’ from one set of values to an ‘output’ in another
For example, a mapping diagram for the function $x + 3$ where $x \geq 3$ could be shown as:

Language of Functions Notes Diagram 2, A Level & AS Level Pure Maths Revision Notes

Worked example

A function is defined as $f (x) = 3 x^{2} - 2 x + 1$ .

(a)

Find

f (7)

The input is

x = 7

, so substitute 7 into the expression everywhere you see an

x

f (7) = 3 {(7)}^{2} - 2 (7) + 1

Calculate.

\begin{array}{rcl} f (7) & = & 3 (49) - 14 + 1 \\ = & 147 - 14 + 1 \end{array}

f (7) = 134

(b)

Find

f (x + 3)

The input is

x = x + 3

so substitute

x + 3

into the expression everywhere you see an

x

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

Expand the brackets and simplify.

$\begin{array}{rcl} f (x + 3) & = & 3 (x^{2} + 6 x + 9) - 2 (x + 3) + 1 \\ = & 3 x^{2} + 18 x + 27 - 2 x - 6 + 1 \\ = & 3 x^{2} + 16 x + 22 \end{array}$

$f (x + 3) = 3 x^{2} + 16 x + 22$

A second function is defined $g : x \mapsto 3 x - 4$ .

(c)

Find the value of

x

for which

g : x \mapsto - 16

Form an equation by setting the function equal to -16.

\begin{array}{rcl} 3 x - 4 & = & - 16 \end{array}

Solve the equation by first adding 4 to both sides, then dividing by 3.

\begin{array}{rcl} 3 x - 4 & = & - 16 \\ 3 x & = & - 12 \\ x & = & - \frac{12}{3} \end{array}

x = - 4

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Domain & Range

How are functions related to graphs?

Functions can be represented as graphs on x and y axes
- The x-axis values are the inputs
- The y-axis values are the outputs
To see what graph to plot, replace f(x) = with y =

DRE Notes fig1, downloadable IGCSE & GCSE Maths revision notes

What is the domain of a function?

The domain of a function is the set of all inputs that the function is allowed to take
Domains can be described in words
- they must refer to x
  - not y or f(x)
- you can use "not equal to" ≠ if needed
- you can use inequality signs if needed
Examples of domains are below:
- f(x) = 3x + 2 takes any x value
  - the domain is "all values of x"
- f(x) = $\frac{1}{x}$ takes any x value except 0 (you cannot divide by 0)
  - the domain is "all values of x except 0", or simply "x ≠ 0"
- f(x) = $\sqrt{x}$ takes any x value that are not negative (you cannot square root a negative)
  - the domain is "x ≥ 0"
- f(x) = x² takes any x value (negative x values are fine as inputs)
  - the domain is "all values of x"
Some domains are restricted by choice
- f(x) = 3x + 2 with the domain 0 < x < 5
  - This question wants to concentrate on that domain only (even though bigger domains exist)
Some domains must exclude certain values (or sets of values)
- f(x) = $\frac{1}{(x - 1) (x + 7)}$ must exclude x = 1 and x = -7 from any domain
  - These two inputs make the function undefined (dividing by zero)
- f(x) = $\sqrt{x - 3}$ must exclude x < 3 from any domain
  - Any input in x < 3 leads to square-rooting a negative

DRE Notes fig2, downloadable IGCSE & GCSE Maths revision notes

What is the range of a function?

The range of a function is the set of all outputs that the function gives out
Ranges can be described in words
- they must refer to f(x)
  - not x or y
- you can use "not equal to" ≠ if needed
- you can use inequality signs if needed
Ranges are influenced by domains
Examples of ranges are below:
- f(x) = 3x + 2 with domain x > 0
  - The range is "f(x) > 2"
  - This is because if the inputs are all greater than 0, the outputs will all be greater than 2
  - This could be seen from a sketch or by substituting inputs of x > 0 into f(x)
- f(x) = x² with domain "all values of x"
  - The range is f(x) ≥ 0
  - This is because all values of x get squared (so no negative outputs are created)

How do I solve problems involving the domain and range?

You need to be able to identify and explain any exclusions in the domain of a function
You need to be able to deduce the range of a function from its expression and domain

You may also be asked to sketch a graph of a function
- This could involve sketching parts of familiar graphs that are restricted because of the domain and exclusions

Examiner Tip

A graph of the function can help “see” both the domain and range of function (a sketch can help if you have not been given a diagram)

Worked example

Two functions are given by

$f (x) = 10 - x g (x) = \frac{1}{2 x - 1}$

(a)

If the domain of function f is

2 < x \leq 4

, find the range.

The domain is the set of inputs
Substitute x = 2 into f(x) to find its output

\begin{array}{rcl} f (2) & = & 10 - 2 \\ = & 8 \end{array}

Substitute x = 4 into f(x) to find its output

\begin{array}{rcl} f (4) & = & 10 - 4 \\ = & 6 \end{array}

Think of f(x) = 10 - x as a graph

the graph of

y = 10 - x

This straight-line graph has a negative gradient
Between x = 2 and x = 4 the graph decreases from a height of 8 to a height of 6
Relate this to outputs

all outputs are between 6 and 8

Write down the range using f(x)
Remember that the inequality is "equal to" at x = 4, f(x) = 6

the range is

6 \leq f (x) < 8

(b)

Write down the value of x that must be excluded from the domain of function g.

An input cannot cause the function to divide by zero
Find out when "dividing by zero" would happen

2 x - 1 = 0

Solve to find this value of x (the one that must be excluded)

2 x = 1 x = \frac{1}{2}

x = \frac{1}{2}

must be excluded

Functions Toolkit (Cambridge O Level Maths)

Revision Note

Introduction to Functions

What is a function?

What does a function look like?

How does a function work?

What is a mapping diagram?

Worked example

Domain & Range

How are functions related to graphs?

What is the domain of a function?

What is the range of a function?

How do I solve problems involving the domain and range?

Examiner Tip

Worked example

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Author: Mark