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 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) \(y = 2x\).