In mathematics, a function is a relation between sets associated with each element of the first set and a single element of the second set. Typical examples are integer functions or real functions.

This function was initially an idealization of how different quantities depend on other quantities. For example, the position of a planet is a function of time. Historically, this concept was developed in the calculation at the end of the seventeenth century and, until the nineteenth century, the functions considered were differentiated (that is, they had a high degree of regularity). The concept of a function was formalized from the point of view of set theory at the end of the nineteenth century, and its scope expanded considerably.

A function is a process or a relation that associates each element x of a set X which is the domain of a function to a single element y of another set Y which is a codomain of the function (although they can to be identical). If the function is called f, this relation is y = f (x) (expressed from x to f). The element x corresponds to the argument corresponding to the value of the function or to the input of the function. The symbol used to represent the entry in the image of the function, output x or f is a function variable (it is often said that f is a function of the variable x).

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Functions are represented only by a set of functions called function graph. (x, f (x)). If the domain and the codomain are a set of real numbers, each of these pairs can be thought of as the Cartesian coordinates of a point in the plane. These sets of points are called function graphs — a general way of describing functions.

Functions are widely used in most fields of science and mathematics. Functions are considered the “centeral objects of investigation” in most areas of mathematics.

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