Function (nonfiction)

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A function f takes an input x, and returns a single output f(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output.

In mathematics, 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 one output.

A function was 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 function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).

A function is uniquely represented by its graph which is the set of all pairs (x, f (x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.

Example

An example is the function that relates each real number x to its square x2.

The output of a function f corresponding to an input x is denoted by f(x) (read "f of x").

In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9.

Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.)

In the News

Fiction cross-reference

Nonfiction cross-reference

  • Algebraic function (nonfiction) - a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.
  • Algorithm (nonfiction)
  • Complex-valued function (nonfiction) - a function whose values are complex numbers. Not to be confused with complex variable function: the domain of a complex-valued function does not necessarily have any structure related to complex numbers.
  • Functional equation (nonfiction) - any equation in which the unknown represents a function. Often, the equation relates the value of a function (or functions) at some point with its values at other points.
  • Golden ratio (nonfiction)
  • Graph of a function (nonfiction) - formally, the set of all ordered pairs (x, f(x)), such that x is in the domain of the function f; in practice, the graphical representation of this set.
  • Holomorphic function (nonfiction) - a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point.
  • Mathematics (nonfiction)

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