Function (mathematics)

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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.

The red curve is the graph of a function fin the Cartesian plane, consisting of all points with coordinates of the form (x, f(x)). The

property of having one output for each input is represented geometrically by the fact that each vertical line (such as the yellow

line through the origin) has exactly one crossing point with the curve.

Function

x f(x)

By domain and codomain

X B X Bn B

X Z X

X R X Rn X

X C X Cn X
 Classes/properties

Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjecti

ve Bijective

Constructions

Restriction Composition Inverse

Generalizations

Partial Multivalued Implicit

In mathematics, a function[1] 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. 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.) The input variable(s) are sometimes referred to as the argument(s) of the
function.
Functions of various kinds are "the central objects of investigation" [2] in most fields of modern mathematics. There
are many ways to describe or represent a function. Some functions may be defined by
a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called
the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for
selected inputs. A function could be described implicitly, for example as the inverseto another function or as a
solution of a differential equation.
In modern mathematics,[3] a function is defined by its set of inputs, called the domain; a set containing the set of
outputs, and possibly additional elements, as members, called its codomain (or target); and the set of all input-
output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the
word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the
function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and
codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this
function is the set of non-negative real numbers. Collections of functions with the same domain and the same
codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real
analysis, complex analysis, and functional analysis.
In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions, in
those cases where the output is a number. Another important operation defined on functions is function
composition, where the output from one function becomes the input to another function.

Contents
[hide]

1Introduction and examples

2Definition
3Notation
4Specifying a function
o 4.1Graph
o 4.2Formulas and algorithms
o 4.3Computability
5Basic properties
 o 5.1Image and preimage
o 5.2Injective and surjective functions
o 5.3Function composition
o 5.4Identity function
o 5.5Empty function
o 5.6Restrictions and extensions
o 5.7Inverse function
6Types of functions
o 6.1Real-valued functions
o 6.2Multivariate functions
o 6.3Further types of functions
7Function spaces
o 7.1Currying
8Variants and generalizations
o 8.1Definitions used in set theory
o 8.2Alternative definition of a function
o 8.3Partial and multi-valued functions
o 8.4Functions with multiple inputs and outputs
8.4.1Binary operations
o 8.5Functors
9See also
10Notes
11References
12Further reading
13External links

Introduction and examples[edit]

A function that associates any of the four colored shapes to its color.

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a
green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and
yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element
in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no
shape that has more than one color. This function will be referred to as the "color-of-the-shape function".
The input to a function is called the argument and the output is called the value. The set of all permitted inputs to
a given function is called the domain of the function, while the set of permissible outputs is called the codomain.
Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of
the five colors. The concept of a function does not require that every possible output is the value of some
argument, e.g. the color blue is not the color of any of the four shapes in X.
A second example of a function is the following: the domain is chosen to be the set of natural numbers (1, 2, 3, 4,
...), and the codomain is the set of integers (..., 3, 2, 1, 0, 1, 2, 3, ...). The function associates to any natural
number n the number 4n. For example, to 1 it associates 3 and to 10 it associates 6.
A third example of a function has the set of polygons as domain and the set of natural numbers as codomain.
The function associates a polygon with its number of vertices. For example, a triangle is associated with the
number 3, a square with the number 4, and so on.
The term range is sometimes used either for the codomain or for the set of all the actual values a function has.

Definition[edit]