Relation under which each value or element in a set of departure (or domain) is associated with one and only one value or element in a set of arrival (or image), according to a rule of correspondence that describes this association.

A function can be defined in extension or intension.

The pairs belonging to a given function can be represented in different ways, such as by an arrow graph or by a graph in a Cartesian plane.

- Example of extensional definition :
*f*= {(*a*, 1), (*b*, 2), (*c*, 1), (*d*, 3), (*e*, 10)}.

- Example of intensional definition :
*f*= { ( x, y) ∈ \(\mathbb{R}\) × \(\mathbb{R}\) | y=2x+5 }.

### Examples

Consider the function f *f* : X → Y : *x* ↦ 2*x* :

- dom(
*f*) = {0, 1, 2, 3,} - ima(
*f*) = {0, 2, 4, 6}

Consider the function *f* : \(\mathbb{R}\) → \(\mathbb{R}\) : *x* ↦ 2*x* + 1 :

- dom(
*f*) = \(\mathbb{R}\) - ima(
*f*) = \(\mathbb{R}\)

### Notation

The function *f* of A toward B under which every element *x* in A is made to correspond to *y* in B so that *y* = *f*(*x)* is noted as:

\(*f* : A → B : x ↦ y = f(x)\)

### Educational Note

It is important to distinguish between the different elements that characterize a function:

- The rule that defines it, literal description or equation;
- Its graph, arrow graph, or Cartesian graph, for example;
- Its pairs, in the case of a binary relation.

That’s why we don’t say: consider the function \(y = 2x\), but rather: consider the function defined by the rule (or the equation) \(y = 2x\).