2.2PureCore

Functions and Mappings

A mapping links inputs to outputs; a function is a special mapping where every input has exactly one output. This lesson nails the language of domain, range, and the one-to-one vs many-to-one distinction.

25 min Functions and Graphs

Edexcel A level Maths: 2.2 Functions and MappingsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Distinguish mappings from functions
Identify one-to-one and many-to-one functions
State the domain and range
Decide whether a mapping is a function

Mappings and functions

A $\textbf{mapping}$ takes inputs to outputs. It is a $\textbf{function}$ only if every input in the domain maps to $\textbf{exactly one}$ output. A mapping that sends one input to two outputs (like $\pm\sqrt{x}$ ) is not a function.

Domain and range

The $\textbf{domain}$ is the set of allowed inputs; the $\textbf{range}$ is the set of outputs produced. Restricting the domain can change the range and even make a non-function into a function.

\text{domain} \xrightarrow{\ f\ } \text{range}

Inputs (domain) map to outputs (range).

Example 1For

f(x) = x^2

with domain all real

x

, state the range.

1Squares are never negative.

Answer

range

f(x) \ge 0

One-to-one vs many-to-one

A function is $\textbf{one-to-one}$ if each output comes from only one input (e.g. $y = x^3$ ), and $\textbf{many-to-one}$ if several inputs give the same output (e.g. $y = x^2$ , where $\pm 2$ both give $4$ ). This matters for inverses later.

Tip — Only one-to-one functions have an inverse — a quick horizontal-line test on the graph decides it.

Formula recap

\text{function: one input} \to \text{one output}

Definition of a function.

\text{domain} = \text{inputs}, \ \text{range} = \text{outputs}

Key vocabulary.

\text{one-to-one vs many-to-one}

Function types.

Common mistakes to avoid

Calling every mapping a function.

A function needs exactly one output per input.

Confusing domain (inputs) with range (outputs).

Domain feeds in; range comes out.

Key takeaways

A function is a mapping with exactly one output per input.
Domain = inputs; range = outputs.
One-to-one (each output once) vs many-to-one (outputs repeat).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A function maps each input to:
- Exactly one output
- Many outputs
- No output
- Two outputs
Show answer and explanation
Answer: Exactly one output
Defining property.
The set of allowed inputs is the:
- Domain
- Range
- Codomain
- Image
Show answer and explanation
Answer: Domain
Inputs = domain.
The range of $f(x) = x^{2}$ (real domain) is:
- $f(x) \ge 0$
- all real numbers
- $f(x) > 0$
- $f(x) \le 0$
Show answer and explanation
Answer: $f(x) \ge 0$
Squares are non-negative.
$y = x^{2}$ is an example of a ___ function.
- Many-to-one
- One-to-one
- One-to-many
- Non-existent
Show answer and explanation
Answer: Many-to-one
2 and −2 both give 4.
Which is one-to-one?
- $y = x^{3}$
- $y = x^{2}$
- $y = |x|$
- $y = \cos x$
Show answer and explanation
Answer: $y = x^{3}$
Each output from one input.
A mapping that gives two outputs for one input is:
- Not a function
- A one-to-one function
- A many-to-one function
- Always invertible
Show answer and explanation
Answer: Not a function
Functions need a single output.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.