Definition of a function

A function f from a set X to a setY is defined by a set G of ordered pairs (x, y), such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G. In other words, for every x in X there is exactly one element y, such that the ordered pair (x, y) belongs to the set of pairs defining the function f.

In this definition, X and Y are respectively called the domain and the codomain of the function f. If (x, y) belongs to the set defining f, then y is the image of x under f, or the value of f applied to the argument x. Especially in the context of numbers, one says also that y is the value of f for the value x of its variable, or, still shorter, y is the value of f of x, denoted as y = f(x).