A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly out output. An example of a commonly used function that related each real number x to its square x^{2}. The output of a function f corresponding to an input x is denoted by f(x). The input variables are often referred to as the arguments of the function.

Functions are very important in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions are defined by a formula or an algorithm that tells how to compute the output for a given input. Some functions are given by a graph of the function. A function can also be described implicitly or as a solution of a differential equation.

The input and output of a function can be expressed as an ordered pair. These pairs are ordered so that the first element is the input and the second is the output. If the input and outputs are real numbers, this ordered pair is viewed as the Cartesian coordinates of a point on a graph of the function.

A function is defined by its set of inputs. These inputs are called the domain. A set containing the outputs is called the codomain or range. The set of all paired inputs and outputs are called the functions graph.

A function f with domain X and codomain Y is commonly denoted by

F: X→Y

The elements of X are called arguments of f. For each argument X there is a corresponding unique Y in the codomain called the function value at x.