What a function is

The concept of a function formalises the idea of a “law” that associates one and only one output with each input.

Definition — Function

Given two non-empty sets $A$ and $B$, a function $f$ from $A$ to $B$ (written $f:A\to B$) is a law that associates with each element $x\in A$ one and only one element $y\in B$, called the image of $x$ under $f$ and denoted $f(x)$.

Remark — The uniqueness of the image

The “one and only one” condition is crucial. A law that associates with $x = 2$ both $y=3$ and $y=5$ is not a function. A law that associates no $y$ with $x = 2$ is not a function on the whole of $A$ (but it may be one on a subset of $A$, which will then be its “natural domain”).

Links

Topics: Functions and properties
Concepts: Function · Image