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 Video by Zeeshan Zamurred Functions and Graphs

Edexcel A level Maths: 2.2 Functions and MappingsWatch the full walkthrough 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 $mapping$ takes inputs to outputs. It is a $function$ only if every input in the domain maps to $exactly one$ output. A mapping that sends one input to two outputs (like $\pm x$ ) is not a function.

Domain and range

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

domain f range

Inputs (domain) map to outputs (range).

1Squares are never negative.

Answerrange

f (x) \geq 0

One-to-one vs many-to-one

A function is $one-to-one$ if each output comes from only one input (e.g. $y = x^{3}$ ), and $many-to-one$ 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

function: one input \to one output

Definition of a function.

domain = inputs, range = outputs

Key vocabulary.

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

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