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 }.


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) = \(\mathbb{R}\)

fonction droite


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) \(y = 2x\).

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