Functions

This lesson covers the formal definition of functions, domain and range, composite functions, and inverse functions — the language and machinery of A-Level mathematics.

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What Is a Function?

A function is a mapping that takes each element from a set (the domain) to exactly one element in another set (the codomain). The set of actual output values is the range.

Notation

f: x ↦ 2x + 1 means “f maps x to 2x + 1”.

Equivalently: f(x) = 2x + 1.

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Domain and Range

Term	Meaning
Domain	The set of allowed input values
Range	The set of all output values

Worked Example 1

f(x) = √(x − 3). Find the domain and range.

Domain: x − 3 ≥ 0 → x ≥ 3, so domain is [3, ∞).

Range: √(x − 3) ≥ 0, so range is [0, ∞).

Worked Example 2

g(x) = 1/(x − 2), x ≠ 2.

Domain: all real numbers except x = 2. Range: all real numbers except y = 0 (since 1/(x − 2) can never equal zero).

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Types of Mapping