# Function

## Function

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 ↦ 2x :

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

Consider the function f : $$\mathbb{R}$$ → $$\mathbb{R}$$ : x ↦ 2x + 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 yf(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) $$y = 2x$$.