Function (mathematics)

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.^[1] The set X is called the domain of the function^[2] and the set Y is called the codomain of the function.^[3]

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept.

A function is often denoted by a letter such as f, g or h. The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x) is denoted by f(x); for example, the value of f at x = 4 is denoted by f(4). Commonly, a specific function is defined by means of an expression depending on x, such as ${\displaystyle f(x)=x^{2}+1;}$ in this case, some computation, called function evaluation, may be needed for deducing the value of the function at a particular value; for example, if ${\displaystyle f(x)=x^{2}+1,}$ then ${\displaystyle f(4)=4^{2}+1=17.}$

Given its domain and its codomain, a function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function.^{[note 1][4]} When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.

Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.^[5]

The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details.

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