Functions: Domain, Range & Mapping

This lesson covers functions, domain, range, and mappings as required by the AQA A-Level Mathematics specification (7357). A solid understanding of functions is essential for the rest of the course — you will need to know what makes a relation a function, how to determine domain and range, and how restricting the domain can make a function invertible.

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What Is a Function?

A function is a mapping that takes each element of a set (called the domain) and assigns it to exactly one element of another set (called the codomain). We write f : A → B to mean "f is a function from set A to set B".

The key requirement is: every input produces exactly one output.

Function vs Relation

A relation is any rule that connects elements of one set to elements of another. Not every relation is a function.

Example: The relation y² = x is not a function of x, because for each positive x there are two values of y (e.g. if x = 4, then y = 2 or y = −2).