View source for Function (mathematics)

{{short description|Mapping that associates a single output value to each input}}
{{Functions}}

In [[mathematics]], a '''function'''<ref group=note>The words '''map''', '''mapping''', '''transformation''', '''correspondence''', and '''operator''' are often used synonymously. {{harvnb |Halmos |1970 |p=30}}.</ref> is a [[binary relation|relation]] between [[set (mathematics)|sets]] that associates to every element of a first set exactly one element of the second set. Typical examples are functions from [[integer]]s to integers or from the [[real number]]s to real numbers.

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. [[History of the function concept|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 function|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<!-- Please, do not link to [[Binary relation]], this is not the technical meaning that is intended--> that associates each element {{mvar|x}} of a [[set (mathematics)|set]] {{mvar|X}},  the ''[[domain of a function|domain]]'' of the function, to a single element {{mvar|y}} of another set {{mvar|Y}} (possibly the same set), the ''[[codomain]]'' of the function. If the function is called {{mvar|f}}, this relation is denoted {{math|1=''y'' = ''f''{{space|hair}}(''x'')}} (which is spoken aloud as {{mvar|f}} of {{mvar|x}}), the element {{mvar|x}} is the ''[[argument of a function|argument]]'' or ''input'' of the function, and {{mvar|y}} is the ''value of the function'', the ''output'', or the ''image'' of {{mvar|x}} by {{mvar|f}}.<ref name=MacLane>{{cite book | last = MacLane | first = Saunders | authorlink = Saunders MacLane | last2 = Birkhoff | first2 = Garrett | author2-link = Garrett Birkhoff | title = Algebra | url = https://archive.org/details/algebra00macl | url-access = registration | publisher = Macmillan | edition = First | year = 1967 | location = New York | pages = [https://archive.org/details/algebra00macl/page/1 1–13] }}</ref> The symbol that is used for representing the input is the [[variable (mathematics)|variable]] of the function (one often says that {{mvar|f}} is a function of the variable {{mvar|x}}).

A function is uniquely represented by the set of all [[pair (mathematics)|pairs]] {{math|(''x'', ''f''{{space|hair}}(''x''))}}, called the [[graph of a function|graph]] of the function.<ref group=note>This definition of "graph" refers to a ''set'' of pairs of objects. Graphs, in the sense of ''diagrams'', are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices).</ref> When the domain and the codomain are sets of real numbers, each such pair may be considered as the [[Cartesian coordinates]] of a point in the plane. The set of these points is called the graph of the function; it is a popular means to illustrate the function.

Functions are widely used in [[science]], and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.{{sfn |Spivak |2008 |p=39}}

[[File:Function machine2.svg|thumb|right|Schematic depiction of a function described metaphorically as a "machine" or "[[black box]]" that for each input yields a corresponding output]]
[[Image:Graph of example function.svg|thumb|right|The red curve is the [[graph of a function]], because any [[Vertical line test|vertical line]] has exactly one crossing point with the curve.]]

[[File:Function color example 3.svg|thumb|A function that associates any of the four colored shapes to its color.]]

==Definition==
{{multiple image| width = 220
| align     = right
| direction = vertical
| image1    = Injection keine Injektion 2a.svg
| caption1  = Diagram of a function, with domain ''X''={1, 2, 3} and codomain ''Y''={A, B, C, D}, which is defined by the set of ordered pairs {(1,D), (2,C), (3,C)}. The image/range is the set {C,D}.
<br /><hr style="height:8pt; visibility:hidden">
----<hr style="height:8pt; visibility:hidden">
|
| image2    = Injection keine Injektion 1.svg
| caption2  = This diagram, representing the set of pairs {(1,D), (2,B), (2,C)}, does ''not'' define a function. One reason is that 2 is the first element in more than one ordered pair, {{nowrap|(2, B)}} and {{nowrap|(2, C)}}, of this set.  Two other reasons, also sufficient by themselves, is that neither 3 nor 4 are first elements (input) of any ordered pair therein.
}}
Intuitively, a function is a process that associates to each element of a set {{math|''X''}} a single element of a set {{math|''Y''}}.

Formally, a function {{math|''f''}} from a set {{math|''X''}} to a set {{math|''Y''}} is defined by a set {{mvar|G}} of ordered pairs {{math|(''x'', ''y'')}} such that {{math|''x'' ∈ ''X''}}, {{math|''y'' ∈ ''Y''}}, and every element of {{math|''X''}} is the first component of exactly one ordered pair in {{mvar|G}}.<ref>{{cite book |last=Hamilton |first=A. G. |title=Numbers, sets, and axioms: the apparatus of mathematics |page=83 |publisher=Cambridge University Press |isbn=978-0-521-24509-8 |url=https://books.google.com/books?id=OXfmTHXvRXMC&pg=PA83&dq=%22function+is+a+relation%22|year=1982 }}</ref><ref group=note>The sets ''X'', ''Y'' are parts of data defining a function; i.e., a function is a set of ordered pairs <math>(x, y)</math> with <math>x \in X, y \in Y</math>, together with the sets ''X'', ''Y'', such that for each <math>x \in X</math>, there is a unique <math>y \in Y</math> with <math>(x, y)</math> in the set.</ref> In other words, for every {{math|''x''}} in {{math|''X''}}, there is exactly one element {{math|''y''}} such that the ordered pair {{math|(''x'', ''y'')}} belongs to the set of pairs defining the function {{math|''f''}}. The set {{mvar|G}} is called the [[graph of a function|graph of the function]]. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called ''[[Map (mathematics)|maps]]'' or ''mappings'', though some authors make some distinction between "maps" and "functions" (see section [[#Map]]).

In the definition of function, {{math|''X''}} and {{math|''Y''}} are respectively called the ''domain'' and the ''codomain'' of the function {{mvar|f}}. If {{math|(''x'', ''y'')}} belongs to the set defining {{mvar|f}}, then {{mvar|y}} is the ''image'' of {{mvar|x}} under {{mvar|f}}, or the ''value'' of {{mvar|f}} applied to the ''argument'' {{mvar|x}}. Especially in the context of numbers, one says also that {{mvar|y}} is the value of {{mvar|f}} for the ''value {{mvar|x}} of its variable'', or, still shorter, {{mvar|y}} is the ''value of'' {{mvar|f}} ''of'' {{mvar|x}}, denoted as {{math|''y'' {{=}} ''f''(''x'')}}.

Two functions {{mvar|f}} and {{mvar|g}} are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, {{math|1=''f'' = ''g''}} if {{math|1=''f''(''x'') = ''g''(''x'')}} for all {{math|''x'' ∈ ''X''}}, where {{math|''f'':''X'' → ''Y''}} and {{math|''g'':''X'' → ''Y''}}.{{sfn |Apostol |1981 |p=35}}{{sfn |Kaplan |1972 |p=25}}<ref group=note>This follows from the [[axiom of extensionality]], which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example, {{cite web |url=https://math.stackexchange.com/q/1403122 |date=August 19, 2015 |title=When do two functions become equal? |work=[[Stack Exchange]] }}</ref>

The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one knows only that the domain is contained in a larger set. Typically, this occurs in [[mathematical analysis]], where "a function {{nowrap|from {{mvar|X}} to {{mvar|Y}} "}} often refers to a function that may have a proper subset of {{mvar|X}} as domain. For example, a "function from the reals to the reals" may refer to a [[real-valued function|real-valued]] function of a [[function of a real variable|real variable]], and this phrase does not mean that the domain of the function is the whole set of the [[real number]]s, but only that the domain is a set of real numbers that contains a non-empty [[open interval]]; such a function is then called a [[partial function]]. For example, if {{mvar|f}} is a function that has the real numbers as domain and codomain, then a function mapping the value {{mvar|x}} to the value <math>g(x)=\tfrac 1{f(x)}</math> is a function {{mvar|g}} from the reals to the reals, whose domain is the set of the reals {{mvar|x}}, such that {{math|''f''(''x'') ≠ 0}}.

The [[range (mathematics)|range]] of a function is the set of the images of all elements in the domain. However, ''range'' is sometimes used as a synonym of codomain, generally in old textbooks.{{citation needed|reason=Give an example reference for both meanings, also to illustrate what 'old' refers to (this changes over time).|date=March 2019}}

===Relational approach===
Any subset of the Cartesian product of two sets <math>X</math> and <math>Y</math> defines a [[binary relation]] <math>R\subseteq X \times Y</math> between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function, given above.

A [[binary relation#univalent|univalent]] or [[binary relation#univalent|functional]] relation is a relation such that
:<math>(x,y)\in R\; \land \;(x,z) \in R \implies y=z</math>
for all <math>x</math> in <math>X</math> and <math>y, z</math> in <math>Y</math>. Univalent relations may be identified with functions with codomain <math>Y,</math> whose domain is a subset of {{mvar|X}}.

A [[total relation]] is a relation such that
:<math>\forall x \in X \;\exists y \in Y : (x,y)\in R.</math>
Formally, functions are identified as relations that are both univalent and total. Univalent relations that are ''not'' total are called [[partial function]]s.

Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is [[injective function|injective]] if the [[converse relation]] <math>R^{\text{T}}\subseteq Y \times X</math> is univalent, where the converse relation is defined as <math>R^{\text{T}} = \{(y,x) \mid (x,y) \in R\}.</math><ref name=RM>[[Gunther Schmidt]]( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60, [[Cambridge University Press]] {{ISBN|978-0-521-76268-7}}  [http://www.cambridge.org/us/academic/subjects/mathematics/logic-categories-and-sets/relational-mathematics?format=HB CUP blurb for ''Relational Mathematics'']</ref>

===As an element of a Cartesian product over a domain ===
The set of all functions from some given domain to a codomain is sometimes identified with the Cartesian product of copies of the codomain, [[index set|indexed]] by the domain. Namely, given sets <math>X</math> and <math>Y,</math> any function <math>f\colon X \to Y</math> is an element of the Cartesian product of copies of <math>Y</math>s over the index set <math>X</math>
:<math>f \in \prod_X Y = Y^X.</math>

Viewing <math>f</math> as tuple with coordinates, then for each <math>x \in X</math>, the <math>x</math>th coordinate of this tuple is the value <math>f(x) \in Y.</math> This reflects the intuition that for each <math>x \in X,</math> the function ''picks'' some element <math>y \in Y,</math> namely, <math>f(x).</math> (This point of view is used for example in the discussion of a [[choice function]].)

Infinite Cartesian products are often simply "defined" as sets of functions.<ref>Halmos, Naive Set Theory, 1968, sect.9 ("Families")</ref>

==Notation==

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which defines the function using an equation that gives the names of the function and the argument explicitly.  This gives rise to a subtle point, often glossed over in elementary treatments of functions: ''functions'' are distinct from their ''values''.  Thus, a function {{math|''f''}} should be distinguished from its value {{math|''f''(''x''<sub>0</sub>)}} at the value {{math|''x''<sub>0</sub>}} in its domain.  To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid appearing pedantic.  However, strictly speaking, it is an [[abuse of notation]] to write "let <math>f\colon \mathbb{R}\to\mathbb{R}</math> be the function {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} ", since {{math|''f''(''x'')}} and {{math|''x''<sup>2</sup>}} should both be understood as the ''value'' of ''f'' at ''x'', rather than the function itself.  Instead, it is correct, though long-winded, to write "let <math>f\colon \mathbb{R}\to\mathbb{R}</math> be the function defined by the equation {{math|1=''f''(''x'') = ''x''<sup>2</sup>,}} valid for all real values of {{math|''x''}} ".  A compact phrasing is "let <math>f\colon \mathbb{R}\to\mathbb{R}</math> with {{math|1=''f''(''x'') = ''x''<sup>2</sup>,}}" where the redundant "be the function" is omitted and, by convention, "for all <math>x</math> in the domain of <math>f</math>" is understood.

This distinction in language and notation becomes important in cases where functions themselves serve as inputs for other functions.  (A function taking another function as an input is termed a ''[[Functional (mathematics)|functional]]''.)  Other approaches to denoting functions, detailed below, avoid this problem but are less commonly used.

===Functional notation===
As first used by [[Leonhard Euler]] in 1734,<ref>{{citation |page=19 |title=Calculus of a Single Variable |author=Ron Larson, Bruce H. Edwards |publisher=Cengage Learning |year=2010 |isbn=978-0-538-73552-0}}</ref> functions are denoted by a symbol consisting generally of a single letter in [[italic font]], most often the lower-case letters {{math|''f'', ''g'', ''h''}}. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). By convention, in this case, a [[roman type]] is used, such as "{{math|sin}}" for the [[sine function]], in contrast to italic font for single-letter symbols.

The notation (read: "{{mvar|y}} equals {{mvar|f}} of {{mvar|x}}")
:<math>y=f(x)</math>
means that the pair {{math|(''x'', ''y'')}} belongs to the set of pairs defining the function {{mvar|f}}. If {{mvar|X}} is the domain of {{mvar|f}}, the set of pairs defining the function is thus, using [[set-builder notation]],
:<math>\{(x, f(x)) : x\in X\}.</math>

Often, a definition of the function is given by what ''f'' does to the explicit argument ''x.''  For example, a function ''f'' can be defined by the equation

:<math>f(x)=\sin(x^2+1)</math>

for all real numbers ''x.''  In this example, ''f'' can be thought of as the [[#Function composition|composite]] of several simpler functions: squaring, adding 1, and taking the sine.  However, only the sine function has a common explicit symbol (sin), while the combination of squaring and then adding 1 is described by the polynomial expression <math>x^2+1</math>.  In order to explicitly reference functions such as squaring or adding 1 without introducing new function names (e.g., by defining function ''g'' and ''h'' by <math>g(x)=x^2</math> and <math>h(x)=x+1</math>), one of the methods below (arrow notation or dot notation) could be used.

Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise.  For example, <math>\sin x</math> can be written instead of <math>\sin(x).</math>

===Arrow notation===
For explicitly expressing domain {{math|''X''}} and the codomain {{math|''Y''}} of a function {{math|''f''}}, the arrow notation is often used (read: {{nowrap|"the function {{math|''f''}} from {{mvar|X}} to {{mvar|Y}}"}} or {{nowrap|"the function {{math|''f''}} mapping elements of  {{mvar|X}} to elements of {{mvar|Y}}"}}):

:<math>f\colon X \to Y</math>
or
:<math>X ~ \stackrel{f}{\to} ~ Y.</math>

This is often used in relation with the arrow notation for elements (read: "{{mvar|f}} maps {{mvar|x}} to {{math|''f''{{space|hair}}(''x'')}}"), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:
:<math>x\mapsto f(x).</math>

For example, if a multiplication is defined on a set {{mvar|X}}, then the [[square function]] <math>\operatorname{sqr}</math> on {{mvar|X}} is unambiguously defined by (read: "the function <math>\operatorname{sqr}</math> from {{mvar|X}} to {{mvar|X}} that maps {{mvar|x}} to {{math|''x'' ⋅ ''x''}}")
:<math>\begin{align}
\operatorname{sqr}\colon X &\to X\\
x &\mapsto x\cdot x,\end{align}</math>

the latter line being more commonly written
:<math>x\mapsto x^2.</math>

Often, the expression giving the function symbol, domain and codomain is omitted.  Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument.  As a common application of the arrow notation, suppose <math>f\colon X\times X\to Y;\;(x,t) \mapsto f(x,t)</math> is a two-argument function, and we want to refer to a [[Partial application|partially applied function]] <math>X\to Y</math> produced by fixing the second argument to the value {{math|''t''<sub>0</sub>}} without introducing a new function name.  The map in question could be denoted <math>x\mapsto f(x,t_0)</math> using the arrow notation for elements. The expression <math>x\mapsto f(x,t_0)</math> (read: "the map taking {{math|''x''}} to <math>f(x,t_0)</math>") represents this new function with just one argument, whereas the expression <math>f(x_0,t_0)</math> refers to the value of the function {{math|''f''}} at the {{nowrap|point <math>(x_0,t_0)</math>.}}

===Index notation===

Index notation is often used instead of functional notation. That is, instead of writing {{math|''f''{{space|hair}}(''x'')}}, one writes <math>f_x.</math>

This is typically the case for functions whose domain is the set of the [[natural number]]s. Such a function is called a [[sequence (mathematics)|sequence]], and, in this case the element <math>f_n</math> is called the {{mvar|n}}th element of sequence.

The index notation is also often used for distinguishing some variables called [[parameter]]s from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem.  For example, the map <math>x\mapsto f(x,t)</math> (see above) would be denoted <math>f_t</math> using index notation, if we define the collection of maps <math>f_t</math> by the formula <math>f_t(x)=f(x,t)</math> for all <math>x,t\in X</math>.

===Dot notation===

In the notation
<math>x\mapsto f(x),</math>
the symbol {{mvar|x}} does not represent any value, it is simply a [[placeholder name|placeholder]] meaning that, if {{mvar|x}} is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, {{mvar|x}} may be replaced by any symbol, often an [[interpunct]] "{{math| ⋅ }}". This may be useful for distinguishing the function {{math|''f''{{space|hair}}(⋅)}} from its value {{math|''f''{{space|hair}}(''x'')}} at {{mvar|x}}.

For example, <math> a(\cdot)^2</math> may stand for the function <math> x\mapsto ax^2</math>, and <math>\textstyle \int_a^{\, (\cdot)} f(u)\,du</math> may stand for a function defined by an integral with variable upper bound: <math>\textstyle x\mapsto \int_a^x f(u)\,du</math>.

=== Specialized notations ===
There are other, specialized notations for functions in sub-disciplines of mathematics.  For example, in [[linear algebra]] and [[functional analysis]], [[linear form]]s and the [[Vector (mathematics and physics)|vectors]] they act upon are denoted using a [[dual pair]] to show the underlying [[Duality (mathematics)|duality]].  This is similar to the use of [[bra–ket notation]] in quantum mechanics.  In [[Mathematical logic|logic]] and the [[theory of computation]], the function notation of [[lambda calculus]] is used to explicitly express the basic notions of function [[Abstraction (computer science)|abstraction]] and [[Function application|application]].  In [[category theory]] and [[homological algebra]], networks of functions are described in terms of how they and their compositions [[Commutative property|commute]] with each other using [[commutative diagram]]s that extend and generalize the arrow notation for functions described above.

== Other terms ==
{| class="wikitable floatright" style= "width: 50%"
!Term
!Distinction from "function"
|-
| rowspan="3" |[[Map (mathematics)|Map/Mapping]]
|None; the terms are synonymous.<ref>{{Cite web|url=http://mathworld.wolfram.com/Map.html|title=Map|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-06-12}}</ref>
|-
|A map can have ''any set'' as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of [[real number|real]] or [[complex number|complex]] numbers.<ref>{{citation|last=Lang|first=Serge|title=Linear Algebra|page=83|year=1971|edition=2nd|publisher=Addison-Wesley}}</ref>
|-
|Alternatively, a map is associated with a ''special structure'' (e.g. by explicitly specifying a structured codomain in its definition). For example, a [[linear map]].<ref>{{cite book|title=Mathematical Analysis|author=T. M. Apostol|publisher=Addison-Wesley|year=1981|page=35}}</ref>
|-
|[[Homomorphism]]
|A function between two [[structure (mathematics)|structures]] of the same type that preserves the operations of the structure (e.g. a [[group homomorphism]]).<ref name=":0">{{Cite web|url=https://ncatlab.org/nlab/show/function|title=function in nLab|website=ncatlab.org|access-date=2019-06-12}}</ref><ref>{{Cite web|url=https://ncatlab.org/nlab/show/homomorphism|title=homomorphism in nLab|website=ncatlab.org|access-date=2019-06-12}}</ref>
|-
|[[Morphism]]
|A generalisation of homomorphisms to any [[Category (mathematics)|category]], even when the objects of the category are not sets (for example, a [[group (mathematics)|group]] defines a category with only one object, which has the elements of the group as morphisms; see {{slink|Category (mathematics)|Examples}} for this example and other similar ones).<ref>{{cite web|url=https://ncatlab.org/nlab/show/morphism|title=morphism|publisher=nLab|access-date=2019-06-12}}</ref><ref name=":0" /><ref>{{Cite web|url=http://mathworld.wolfram.com/Morphism.html|title=Morphism|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-06-12}}</ref>
|}

===Map===
A function is often also called a '''map''' or a '''mapping''', but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. [[maps of manifolds]]). In particular ''map'' is often used in place of ''homomorphism'' for the sake of succinctness (e.g., [[linear map]] or ''map from {{mvar|G}} to {{mvar|H}}'' instead of ''[[group homomorphism]] from {{mvar|G}} to {{mvar|H}}''). Some authors<ref>{{cite book|author=T. M. Apostol|title=Mathematical Analysis|year=1981|publisher=Addison-Wesley|page=35}}</ref> reserve the word ''mapping'' for the case where the structure of the codomain belongs explicitly to the definition of the function. 

Some authors, such as [[Serge Lang]],<ref>{{citation|first=Serge|last=Lang|title=Linear Algebra|edition=2nd|year=1971|page=83|publisher=Addison-Wesley}}</ref> use "function" only to refer to maps for which the [[codomain]] is a subset of the [[real number|real]] or [[complex number|complex]] numbers, and use the term ''mapping'' for more general functions.

In the theory of [[dynamical system]]s, a map denotes an [[Discrete-time dynamical system|evolution function]] used to create [[Dynamical system#Maps|discrete dynamical systems]]. See also [[Poincaré map]].

Whichever definition of ''map'' is used, related terms like ''[[Domain (mathematics)|domain]]'', ''[[codomain]]'', ''[[Injective function|injective]]'', ''[[Continuous function|continuous]]'' have the same meaning as for a function.

===Morphism===
Because the term "map" is synonymous with "[[morphism (category theory)|morphism]]" in category theory, the term "map" can in particular emphasize the aspect that a function is a morphism in the [[category of sets]]: in the informal definition of a function <math>f\colon X\to Y</math>, <math>f</math> is a subset of <math>X\times Y</math> consisting of all the pairs <math>(x,f(x))</math> for <math>x\in X</math>. In this sense, the function doesn't capture the information that <math>Y</math> is the codomain; only the range <math>f(X)</math> is determined by the function.

==Specifying a function==
Given a function <math>f</math>, by definition, to each element <math>x</math> of the domain of the function <math>f</math>, there is a unique element associated to it, the value <math>f(x)</math> of <math>f</math> at <math>x</math>. There are several ways to specify or describe how <math>x</math> is related to <math>f(x)</math>, both explicitly and implicitly. Sometimes, a theorem or an [[axiom]] asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function <math>f</math>.

===By listing function values===
On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. E.g., if <math>A = \{ 1, 2, 3 \}</math>, then one can define a function <math>f\colon A \to \mathbb{R}</math> by <math>f(1) = 2, f(2) = 3, f(3) = 4.</math>

===By a formula===
Functions are often defined by a [[closed-form expression|formula]] <!-- "closed-form expression" is too technical here-->that describes a combination of [[arithmetic operations]] and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
For example, in the above example, <math>f</math> can be defined by the formula <math>f(n) = n+1</math>, for <math>n\in\{1,2,3\}</math>.

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the [[zero of a function|zeros]] of auxiliary functions. Similarly, if [[square root]]s occur in the definition of a function from <math>\mathbb{R}</math> to <math>\mathbb{R},</math> the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

For example, <math>f(x)=\sqrt{1+x^2}</math> defines a function <math>f\colon \mathbb{R} \to \mathbb{R}</math> whose domain is <math>\mathbb{R},</math> because <math>1+x^2</math> is always positive if {{mvar|x}} is a real number. On the other hand, <math>f(x)=\sqrt{1-x^2}</math> defines a function from the reals to the reals whose domain is reduced to the interval {{math|[–1, 1]}}. (In old texts, such a domain was called the ''domain of definition'' of the function.)

Functions are often classified by the nature of formulas that can that define them:
*A [[quadratic function]] is a function that may be written <math>f(x)=ax^2+bx+c,</math> where {{math|''a'', ''b'', ''c''}} are [[constant (mathematics)|constants]].
*More generally, a [[polynomial function]] is a function that can be defined by a formula involving only additions, subtractions, multiplications, and [[exponentiation]] to nonnegative integers. For example, <math>f(x)=x^3-3x-1,</math> and <math>f(x)=(x-1)(x^3+1) +2x^2 -1.</math>
*A [[rational function]] is the same, with divisions also allowed, such as <math>f(x)=\frac{x-1}{x+1},</math> and <math>f(x)=\frac 1{x+1}+\frac 3x-\frac 2{x-1}.</math>
*An [[algebraic function]] is the same, with [[nth root|{{mvar|n}}th roots]] and [[zero of a function|roots of polynomials]] also allowed.
*An [[elementary function]]<ref group=note>Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.</ref> is the same, with [[logarithm]]s and [[exponential functions]] allowed.

===Inverse and implicit functions===
A function <math>f\colon X\to Y,</math> with domain {{mvar|X}} and codomain {{mvar|Y}}, is [[bijective]], if for every {{mvar|y}} in {{mvar|Y}}, there is one and only one element {{mvar|x}} in {{mvar|X}} such that {{math|1=''y'' = ''f''(''x'')}}. In this case, the [[inverse function]] of {{mvar|f}} is the function <math>f^{-1}\colon Y \to X</math> that maps <math>y\in Y</math> to the element <math>x\in X</math> such that {{math|1=''y'' = ''f''(''x'')}}. For example, the [[natural logarithm]] is a bijective function from the positive real numbers to the real numbers. It has these an inverse, called the [[exponential function]] that maps the real numbers onto the positive numbers.

If a function <math>f\colon X\to Y</math> is not bijective, it may occur that one can select subsets <math>E\subseteq X</math> and <math>F\subseteq Y</math> such that the [[restriction of a function|restriction]] of {{mvar|f}} to {{mvar|E}} is a bijection from {{mvar|E}} to {{mvar|F}}, and has thus an inverse. The [[inverse trigonometric functions]] are defined this way. For example, the [[cosine function]] induces, by restriction, a bijection from the [[interval (mathematics)|interval]] {{math|[0, ''&pi;'']}} onto the interval {{math|[–1, 1]}}, and its inverse function, called [[arccosine]], maps {{math|[–1, 1]}} onto {{math|[0, ''&pi;'']}}. The other inverse trigonometric functions are defined similarly.

More generally, given a [[binary relation]] {{mvar|R}} between two sets {{mvar|X}} and {{mvar|Y}}, let {{mvar|E}} be a subset of {{mvar|X}} such that, for every <math>x\in E,</math> there is some <math>y\in Y</math> such that {{math|''x R y''}}. If one has a criterion allowing selecting such an {{mvar|y}} for every <math>x\in E,</math> this defines a function <math>f\colon E\to Y,</math> called an [[implicit function]], because it is implicitly defined by the relation {{mvar|R}}.

For example, the equation of the [[unit circle]] <math>x^2+y^2=1</math> defines a relation on real numbers. If {{math|–1 < ''x'' < 1}} there are two possible values of {{mvar|y}}, one positive and one negative. For {{math|1=''x'' = ± 1}}, these two values become both equal to 0. Otherwise, there is no possible value of {{mvar|y}}. This means that the equation defines two implicit functions with domain {{math|[–1, 1]}} and respective codomains {{math|[0, +∞)}} and {{math|(–∞, 0]}}.

In this example, the equation can be solved in {{mvar|y}}, giving <math>y=\pm \sqrt{1-x^2},</math> but, in more complicated examples, this is impossible. For example, the relation <math>y^5+x+1=0</math> defines {{mvar|y}} as an implicit function of {{mvar|x}}, called the [[Bring radical]], which has <math>\mathbb R</math> as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and [[nth root|{{mvar|n}}th roots]].

The [[implicit function theorem]] provides mild [[differentiability]] conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

===Using differential calculus===

Many functions can be defined as the [[antiderivative]] of another function. This is the case of the [[natural logarithm]], which is the antiderivative of {{math|1/''x''}} that is 0 for {{math|1=''x'' = 1}}. Another common example is the [[error function]].

More generally, many functions, including most [[special function]]s, can be defined as solutions of [[differential equation]]s. The simplest example is probably the [[exponential function]], which can be defined as the unique function that is equal to its derivative and takes the value 1 for {{math|1=''x'' = 0}}.

[[Power series]] can be used to define functions on the domain in which they converge. For example, the [[exponential function]] is given by <math>e^x = \sum_{n=0}^{\infty} {x^n \over n!}</math>. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its [[Taylor series]] in some interval, this power series allows immediately enlarging the domain to a subset of the [[complex number]]s, the [[disc of convergence]] of the series. Then [[analytic continuation]] allows enlarging further the domain for including almost the whole [[complex plane]]. This process is the method that is generally used for defining the [[logarithm]], the [[exponential function|exponential]] and the [[trigonometric functions]] of a complex number.

===By recurrence ===
{{main|Recurrence relation}}
Functions whose domain are the nonnegative integers, known as [[sequence]]s, are often defined by [[recurrence relation]]s.

The [[factorial]] function on the nonnegative integers (<math>n\mapsto n!</math>) is a basic example, as it can be defined by the recurrence relation
:<math>n!=n(n-1)!\quad\text{for}\quad n>0,</math>
and the initial condition
:<math>0!=1.</math>

==Representing a function==

A [[Graph of a function|graph]] is commonly used to give an intuitive picture of a function.  As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by [[bar chart]]s.

===Graphs and plots===
{{main|Graph of a function}}
[[File:Motor vehicle deaths in the US.svg|thumb|The function mapping each year to its US motor vehicle death count, shown as a [[line chart]]]]
[[File:Motor vehicle deaths in the US histogram.svg|thumb|The same function, shown as a bar chart]]

Given a function <math>f\colon X\to Y,</math> its ''graph'' is, formally, the set

:<math>G=\{(x,f(x)) : x\in X\}.</math>

In the frequent case where {{mvar|X}} and {{mvar|Y}} are subsets of the [[real number]]s (or may be identified with such subsets, e.g. [[interval (mathematics)|intervals]]), an element <math>(x,y)\in G</math> may be identified with a point having coordinates {{math|''x'', ''y''}} in a 2-dimensional coordinate system, e.g. the [[Cartesian plane]]. Parts of this may create a [[Plot (graphics)|plot]] that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the ''graph of the function''. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the [[square function]]

:<math>x\mapsto x^2,</math>

consisting of all points with coordinates <math>(x, x^2)</math> for <math>x\in \R,</math> yields, when depicted in Cartesian coordinates, the well known [[parabola]]. If the same quadratic function <math>x\mapsto x^2,</math> with the same formal graph, consisting of pairs of numbers, is plotted instead in [[polar coordinates]] <math>(r,\theta) =(x,x^2),</math> the plot obtained is [[Fermat's spiral]].

===Tables===
{{Main|Mathematical table}}
A function can be represented as a table of values.  If the domain of a function is finite, then the function can be completely specified in this way.  For example, the multiplication function <math>f\colon\{1,\ldots,5\}^2 \to \mathbb{R}</math> defined as <math>f(x,y)=xy</math> can be represented by the familiar [[multiplication table]]

{| class="wikitable" style="text-align: center;"
! {{diagonal split header|{{mvar|x}}|{{mvar|y}}}}
! 1  !! 2  !! 3  !! 4  !! 5
|-
! 1
| 1  || 2  || 3  || 4 || 5
|-
! 2
| 2  || 4  ||6 || 8  || 10
|-
! 3
| 3  || 6  || 9  || 12 || 15
|-
! 4
| 4  || 8  || 12 || 16 || 20
|-
! 5
| 5  || 10 || 15 || 20 || 25
|}

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain.  If an intermediate value is needed, [[interpolation]] can be used to estimate the value of the function.  For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

{| class="wikitable" style="text-align: center;"
! {{mvar|x}} !! {{math|sin ''x''}}
|-
|1.289 || 0.960557
|-
|1.290 || 0.960835
|-
|1.291 || 0.961112
|-
|1.292 || 0.961387
|-
|1.293 || 0.961662
|}

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

===Bar chart===
{{main|Bar chart}}
Bar charts are often used for representing functions whose domain is a finite set, the [[natural number]]s, or the [[integer]]s. In this case, an element {{mvar|x}} of the domain is represented by an [[interval (mathematics)|interval]] of the {{mvar|x}}-axis, and the corresponding value of the function, {{math|''f''(''x'')}}, is represented by a [[rectangle]] whose base is the interval corresponding to {{mvar|x}} and whose height is {{math|''f''(''x'')}} (possibly negative, in which case the bar extends below the {{mvar|x}}-axis).

==General properties==

This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.

===Standard functions===
There are a number of standard functions that occur frequently:{{anchor|Empty function}}
* For every set {{mvar|X}}, there is a unique function, called the '''empty function''' from the [[empty set]] to {{mvar|X}}.  The existence of the empty function from the empty set to itself is required for the [[category of sets]] to be a category &ndash; in a category, each object must have an "identity morphism", and the empty function serves as the identity for the empty set.  The existence of a unique empty function from the empty set to every set {{mvar|A}} means that the empty set is an [[initial object]] in the category of sets. In terms of cardinal arithmetic, it means that {{math|1=''k''<sup>0</sup> = 1}} for every [[cardinal number]] {{mvar|k}}.
* For every set {{mvar|X}} and every [[singleton set]] {{math|{{mset|''s''}}}}, there is a unique function from {{mvar|X}} to {{math|{{mset|''s''}}}}, which maps every element of {{mvar|X}} to {{mvar|s}}. This is a surjection (see below) unless {{mvar|X}} is the empty set.
* Given a function <math>f\colon X\to Y,</math> the canonical surjection of {{mvar|f}} onto its image <math>f(X)=\{f(x)\mid x\in X\}</math> is the function from {{mvar|X}} to {{math|''f''(''X'')}} that maps {{mvar|x}} to {{math|''f''(''x'')}}.
* For every [[subset]] {{mvar|A}} of a set {{mvar|X}}, the [[inclusion map]] of {{mvar|A}} into {{mvar|X}} is the injective (see below) function that maps every element of {{mvar|A}} to itself.
* The [[identity function]] on a set {{mvar|X}}, often denoted by {{math|id<sub>''X''</sub>}}, is the inclusion of {{mvar|X}} into itself.

===Function composition===
{{Main|Function composition}}

Given two functions <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> such that the domain of {{mvar|g}} is the codomain of {{mvar|f}}, their ''composition'' is the function <math>g \circ f\colon X \rightarrow Z</math> defined by
:<math>(g \circ f)(x) = g(f(x)).</math>

That is, the value of <math>g \circ f</math> is obtained by first applying {{math|''f''}} to {{math|''x''}} to obtain {{math|1=''y'' =''f''(''x'')}} and then applying {{math|''g''}} to the result {{mvar|y}} to obtain {{math|1=''g''(''y'') = ''g''(''f''(''x''))}}. In the notation the function that is applied first is always written on the right.

The composition <math>g\circ f</math> is an [[operation (mathematics)|operation]] on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both <math>g \circ f</math> and <math>f \circ g</math> satisfy these conditions, the composition is not necessarily [[commutative property|commutative]], that is, the functions <math>g \circ f</math> and <math> f \circ g</math> need not be equal, but may deliver different values for the same argument. For example, let {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} and {{math|1=''g''(''x'') = ''x'' + 1}}, then <math>g(f(x))=x^2+1</math> and <math> f(g(x)) = (x+1)^2</math> agree just for <math>x=0.</math>

The function composition is [[associative property|associative]] in the sense that, if one of <math>(h\circ g)\circ f</math> and <math>h\circ (g\circ f)</math> is defined, then the other is also defined, and they are equal. Thus, one writes
:<math>h\circ g\circ f = (h\circ g)\circ f = h\circ (g\circ f).</math>

The [[identity function]]s <math>\operatorname{id}_X</math> and <math>\operatorname{id}_Y</math> are respectively a [[right identity]] and a [[left identity]] for functions from {{mvar|X}} to {{mvar|Y}}. That is, if {{mvar|f}} is a function with domain {{mvar|X}}, and codomain {{mvar|Y}}, one has
<math>f\circ \operatorname{id}_X = \operatorname{id}_Y \circ f = f.</math>

<gallery widths="250" heights="300">
File:Function machine5.svg|A composite function ''g''(''f''(''x'')) can be visualized as the combination of two "machines".
File:Example for a composition of two functions.svg|A simple example of a function composition
File:Compfun.svg|Another composition. In this example, {{math|1=(''g'' ∘ ''f'' )(c) = #}}.
</gallery>

===Image and preimage===
{{Main|Image (mathematics)}}
Let <math>f\colon X\to Y.</math> The ''image'' by {{mvar|f}} of an element {{mvar|x}} of the domain {{mvar|X}} is {{math|''f''(''x'')}}. If {{math|''A''}} is any subset of {{math|''X''}}, then the ''image'' of {{mvar|A}} by {{mvar|f}}, denoted {{math|''f''(''A'')}} is the subset of the codomain {{math|''Y''}} consisting of all images of elements of {{mvar|A}}, that is,
:<math>f(A)=\{f(x)\mid x\in A\}.</math>

The ''image'' of {{math|''f''}} is the image of the whole domain, that is {{math|''f''(''X'')}}. It is also called the [[range (mathematics)|range]] of {{mvar|f}}, although the term may also refer to the codomain.<ref name = "standard">''Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology'', p. 15.  ISO 80000-2 (ISO/IEC 2009-12-01)</ref>

On the other hand, the ''[[inverse image]]'', or ''[[preimage]]'' by {{mvar|f}} of a subset {{math|''B''}} of the codomain {{math|''Y''}} is the subset of the domain {{math|''X''}} consisting of all elements of {{math|''X''}} whose images belong to {{math|''B''}}. It is denoted by <math>f^{-1}(B).</math>  That is
:<math>f^{-1}(B) = \{x \in X \mid f(x) \in B\}.</math>
For example, the preimage of {4, 9} under the [[square function]] is the set {−3,−2,2,3}.

By definition of a function, the image of an element {{math|''x''}} of the domain is always a single element of the codomain. However, the preimage of a single element {{mvar|y}}, denoted <math>f^{-1}(x),</math> may be [[empty set|empty]] or contain any number of elements. For example, if {{mvar|f}} is the function from the integers to themselves that maps every integer to 0, then <math>f^{-1}(0) = \mathbb{Z}</math>.

If <math>f\colon X\to Y</math> is a function, {{math|''A''}} and {{math|''B''}} are subsets of {{math|''X''}}, and {{math|''C''}} and {{math|''D''}} are subsets of {{math|''Y''}}, then one has the following properties:
* <math>A\subseteq B \Longrightarrow f(A)\subseteq f(B)</math>
* <math>C\subseteq D \Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)</math>
* <math>A \subseteq f^{-1}(f(A))</math>
* <math>C \supseteq f(f^{-1}(C))</math>
* <math>f(f^{-1}(f(A)))=f(A)</math>
* <math>f^{-1}(f(f^{-1}(C)))=f^{-1}(C)</math>

The preimage by {{mvar|f}} of an element {{mvar|y}} of the codomain is sometimes called, in some contexts, the [[fiber (mathematics)|fiber]] of {{math|''y''}} under {{mvar|''f''}}.

If a function {{mvar|f}} has an inverse (see below), this inverse is denoted <math>f^{-1}.</math> In this case <math>f^{-1}(C)</math> may denote either the image by <math>f^{-1}</math> or the preimage by {{mvar|f}} of {{mvar|C}}. This is not a problem, as these sets are equal. The notation <math>f(A)</math> and <math>f^{-1}(C)</math> may be ambiguous in the case of sets that contain some subsets as elements, such as <math>\{x, \{x\}\}.</math> In this case, some care may be needed, for example, by using square brackets <math>f[A], f^{-1}[C]</math> for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

===Injective, surjective and bijective functions===
{{Unreferenced section|date=August 2019}}
Let <math>f\colon X\to Y</math> be a function.

The function {{mvar|f}} is ''[[injective function|injective]]'' (or ''one-to-one'', or is an ''injection'') if {{math|''f''(''a'') ≠ ''f''(''b'')}} for any two different elements {{math|''a''}} and {{mvar|''b''}} of {{mvar|X}}. Equivalently, {{mvar|f}} is injective if, for any <math>y\in Y,</math> the preimage <math>f^{-1}(y)</math> contains at most one element. An empty function is always injective. If {{mvar|X}} is not the empty set, and if, as usual, [[Zermelo–Fraenkel set theory]] is assumed, then {{mvar|f}} is injective if and only if there exists a function <math>g\colon Y\to X</math> such that <math>g\circ f=\operatorname{id}_X,</math> that is, if {{mvar|f}} has a [[left inverse function|left inverse]]. If {{mvar|f}} is injective, for defining {{mvar|g}}, one chooses an element <math>x_0</math> in {{mvar|X}} (which exists as {{mvar|X}} is supposed to be nonempty),<ref group=note>The [[axiom of choice]] is not needed here, as the choice is done in a single set.</ref>  and one defines {{mvar|g}} by <math>g(y)=x</math> if <math>y=f(x),</math> and <math>g(y)=x_0</math>, if <math>y\not\in f(X).</math>

The function {{mvar|f}} is ''[[surjective]]'' (or ''onto'', or is a ''surjection'') if the range equals the codomain, that is, if {{math|1=''f''(''X'') = ''Y''}}. In other words, the preimage <math>f^{-1}(y)</math> of every <math>y\in Y</math> is nonempty. If, as usual, the axiom of choice is assumed, then {{mvar|f}} is surjective if and only if there exists a function <math>g\colon Y\to X</math> such that <math>f\circ g=\operatorname{id}_Y,</math> that is, if {{mvar|f}} has a [[left inverse function|right inverse]]. The axiom of choice is needed, because, if {{mvar|f}} is surjective, one defines {{mvar|g}} by <math>g(y)=x,</math> where <math>x</math> is an ''arbitrarily chosen'' element of <math>f^{-1}(y).</math>

The function {{mvar|f}} is ''[[bijective]]'' (or is ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective. That is {{mvar|f}} is bijective if, for any <math>y\in Y,</math> the preimage <math>f^{-1}(y)</math> contains exactly one element. The function {{mvar|f}} is bijective if and only if it admits an [[inverse function]], that is a function <math>g\colon Y\to X</math> such that <math>g\circ f=\operatorname{id}_X,</math> and <math>f\circ g=\operatorname{id}_Y.</math> (Contrarily to the case of surjections, this does not require the axiom of choice.)

Every function <math>f\colon X\to Y</math> may be [[factorization|factorized]] as the composition {{math|''i'' ∘ ''s''}} of a surjection followed by an injection, where {{mvar|s}} is the canonical surjection of {{mvar|X}} onto {{math|''f''(''X'')}}, and {{mvar|i}} is the canonical injection of {{math|''f''(''X'')}} into {{mvar|Y}}. This is the ''canonical factorization'' of {{mvar|f}}.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the [[Nicolas Bourbaki|Bourbaki group]] and imported into English.  As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function.  Also, the statement "{{math|''f''}} maps {{math|''X''}} ''onto'' {{math|''Y''}}" differs from "{{math|''f''}}  maps {{math|''X''}} ''into'' {{math|''B''}}" in that the former implies that {{math|''f''}} is surjective, while the latter makes no assertion about the nature of {{math|''f''}} the mapping.  In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.

===Restriction and extension{{anchor|Restrictions and extensions}}===<!-- This section is linked from [[Subgroup]], [[Restriction]], [[Quadratic form]] -->
{{main|Restriction (mathematics)}}
If <math>f\colon X \to Y</math> is a function and ''S'' is a subset of ''X'', then the ''restriction'' of <math>f</math> to ''S'', denoted <math>f|_S</math>, is the function from ''S'' to ''Y'' defined by
:<math>f|_S(x) = f(x)</math>
for all ''x'' in ''S''. Restrictions can be used to define partial inverse functions: if there is a subset ''S'' of the domain of a function <math>f</math> such that <math>f|_S</math> is injective, then the canonical surjection of <math>f|_S</math> onto its image <math>f|_S(S) = f(S)</math> is a bijection, and thus has an inverse function from <math>f(S)</math> to ''S''. One application is the definition of [[inverse trigonometric functions]]. For example, the [[cosine]] function is injective when restricted to the [[interval (mathematics)|interval]] {{math|[0, ''&pi;'']}}. The image of this restriction is the interval {{math|[–1, 1]}}, and thus the restriction has an inverse function from {{math|[–1, 1]}} to {{math|[0, ''&pi;'']}}, which is called [[arccosine]] and is denoted {{math|arccos}}.

Function restriction may also be used for "gluing" functions together. Let <math>\textstyle X=\bigcup_{i\in I}U_i</math> be the decomposition of {{mvar|X}} as a [[set union|union]] of subsets, and suppose that a function <math>f_i\colon U_i \to Y</math> is defined on each <math>U_i</math> such that for each pair <math>i, j</math> of indices, the restrictions of <math>f_i</math> and <math>f_j</math> to <math>U_i \cap U_j</math> are equal. Then this defines a unique function <math>f\colon X \to Y</math> such that <math>f|_{U_i} = f_i</math> for all {{mvar|i}}. This is the way that functions on [[manifold]]s are defined.

An ''extension'' of a  function {{mvar|f}} is a function {{mvar|g}} such that {{mvar|f}} is a restriction of {{mvar|g}}. A typical use of this concept is the process of [[analytic continuation]], that allows extending functions whose domain is a small part of the [[complex plane]] to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying [[homography|homographies]] of the [[real line]]. A ''homography'' is a function <math>h(x)=\frac{ax+b}{cx+d}</math> such that {{math|''ad'' – ''bc'' ≠ 0}}. Its domain is the set of all [[real number]]s different from <math>-d/c,</math> and its image is the set of all real numbers different from <math>a/c.</math> If one extends the real line to the [[projectively extended real line]] by including {{math|∞}}, one may extend {{mvar|h}} to a bijection from the extended real line to itself by setting <math>h(\infty)=a/c</math> and <math>h(-d/c)=\infty</math>.

==Multivariate function {{anchor|MULTIVARIATE_FUNCTION}}==
[[File:Binary operations as black box.svg|thumb|A binary operation is a typical example of a bivariate, function which assigns to each pair <math>(x, y)</math> the result <math>x\circ y</math>.]]
A '''multivariate function''', or '''function of several variables''' is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed.

More formally, a function of {{mvar|n}} variables is a function whose domain is a set of [[tuple|{{mvar|n}}-tuples]].
For example, multiplication of [[integer]]s is a function of two variables, or '''bivariate function''', whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every [[binary operation]]. More generally, every [[mathematical operation]] is defined as a multivariate function.

The [[Cartesian product]] <math>X_1\times\cdots\times X_n</math> of {{mvar|n}} sets <math>X_1, \ldots, X_n</math> is the set of all {{mvar|n}}-tuples <math>(x_1, \ldots, x_n)</math> such that <math>x_i\in X_i</math> for every {{mvar|i}} with <math>1 \leq i \leq n</math>. Therefore, a function of {{mvar|n}} variables is a function
:<math>f\colon U\to Y,</math>
where the domain {{mvar|U}} has the form
:<math>U\subseteq X_1\times\cdots\times X_n.</math>
When using function notation, one usually omits the parentheses surrounding tuples, writing <math>f(x_1,x_2)</math> instead of <math>f((x_1,x_2)).</math>

In the case where all the <math>X_i</math> are equal to the set <math>\R</math> of [[real number]]s, one has a [[function of several real variables]]. If the <math>X_i</math> are equal to the set <math>\C</math> of [[complex number]]s, one has a [[function of several complex variables]].

It is common to also consider functions whose codomain is a product of sets. For example, [[Euclidean division]] maps every pair {{math|(''a'', ''b'')}} of integers with {{math|''b'' ≠ 0}} to a pair of integers called the ''quotient'' and the ''remainder'':
:<math>\begin{align}
  \text{Euclidean division}\colon\quad \Z\times (\Z\setminus \{0\}) &\to \Z\times\Z\\
  (a,b) &\mapsto (\operatorname{quotient}(a,b),\operatorname{remainder}(a,b)).
 \end{align}</math>
The codomain may also be a [[vector space]]. In this case, one talks of a [[vector-valued function]]. If the domain is contained in a [[Euclidean space]], or more generally a [[manifold]], a vector-valued function is often called a [[vector field]].

==In calculus==

The idea of function, starting in the 17th century, was fundamental to the new [[infinitesimal calculus]] (see [[History of the function concept]]). At that time, only [[real-valued function|real-valued]] functions of a [[function of a real variable|real variable]] were considered, and all functions were assumed to be [[smooth function|smooth]]. But the definition was soon extended to [[#Multivariate function|functions of several variables]] and to [[functions of a complex variable]].  In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics.  In introductory [[calculus]], when the word ''function'' is used without qualification, it means a real-valued function of a single real variable.  The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as [[real analysis]] and [[complex analysis]].

===Real function===
{{see also|Real analysis}}
[[File:Gerade.svg|thumb|right|Graph of a linear function]]
[[File:Polynomialdeg2.svg|thumb|right|Graph of a polynomial function, here a quadratic function.|right]]
[[File:Sine cosine one period.svg|thumb|right|Graph of two trigonometric functions: [[sine]] and [[cosine]].]]
A ''real function'' is a [[real-valued function|real-valued]] function of a [[function of a real variable|real variable]], that is, a function whose codomain is the [[real number|field of real numbers]] and whose domain is a set of [[real number]]s that contains an [[interval (mathematics)|interval]]. In this section, these functions are simply called ''functions''.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are [[continuous function|continuous]], [[differentiable function|differentiable]], and even [[analytic function|analytic]]. This regularity insures that these functions can be visualized by their [[#graph|graphs]]. In this section, all functions are differentiable in some interval.

Functions enjoy [[pointwise operation]]s, that is, if {{mvar|f}} and {{mvar|g}} are functions, their sum, difference and product are functions defined  by
:<math>\begin{align}
(f+g)(x)&=f(x)+g(x)\\
(f-g)(x)&=f(x)-g(x)\\
(f\cdot g)(x)&=f(x)\cdot g(x)\\
\end{align}.</math>
The domains of the resulting functions are the [[set intersection|intersection]] of the domains of {{mvar|f}} and {{mvar|g}}. The quotient of two functions is defined similarly by
:<math>\frac fg(x)=\frac{f(x)}{g(x)},</math>
but the domain of the resulting function is obtained by removing the [[zero of a function|zeros]] of {{mvar|g}} from the intersection of the domains of {{mvar|f}} and {{mvar|g}}.

The [[polynomial function]]s are defined by [[polynomial]]s, and their domain is the whole set of real numbers. They include [[constant function]]s, [[linear function]]s and [[quadratic function]]s. [[Rational function]]s are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid [[division by zero]].  The simplest rational function is the function <math>x\mapsto \frac 1x,</math> whose graph is a [[hyperbola]], and whose domain is the whole [[real line]] except for 0.

The [[derivative]] of a real differentiable function is a real function. An [[antiderivative]] of a continuous real function is a real function that is differentiable in any [[open interval]] in which the original function is continuous. For example, the function <math>x\mapsto\frac 1x</math> is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for {{math|1=''x'' = 1}}, is a differentiable function called the [[natural logarithm]].

A real function {{mvar|f}} is [[monotonic function|monotonic]] in an interval if the sign of <math>\frac{f(x)-f(y)}{x-y}</math> does not depend of the choice of {{mvar|x}} and {{mvar|y}} in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function {{mvar|f}} is monotonic in an interval {{mvar|I}}, it has an [[inverse function]], which is a real function with domain {{math|''f''(''I'')}} and image {{mvar|I}}. This is how [[inverse trigonometric functions]] are defined in terms of [[trigonometric functions]], where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a [[bijection]] between the real numbers and the positive real numbers. This inverse is the [[exponential function]].

Many other real functions are defined either by the [[implicit function theorem]] (the inverse function is a particular instance) or as solutions of [[differential equation]]s. For example, the [[sine]] and the [[cosine]] functions are the solutions of the [[linear differential equation]]
:<math>y''+y=0</math>
such that
:<math>\sin 0=0, \quad \cos 0=1, \quad\frac{\partial \sin x}{\partial x}(0)=1, \quad\frac{\partial \cos x}{\partial x}(0)=0.</math>

===Vector-valued function===
{{main|Vector-valued function|Vector field}}

When the elements of the codomain of a function are [[vector (mathematics and physics)|vectors]], the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its [[velocity]] vector is a vector-valued function.

Some vector-valued functions are defined on a subset of <math>\mathbb{R}^n</math> or other spaces that share geometric or [[topological]] properties of <math>\mathbb{R}^n</math>, such as [[manifolds]]. These vector-valued functions are given the name ''vector fields''.

==Function space==
{{Main|Function space|Functional analysis}}
In [[mathematical analysis]], and more specifically in [[functional analysis]], a '''function space''' is a set of [[scalar-valued function|scalar-valued]] or [[vector-valued function]]s, which share a specific property and form a [[topological vector space]]. For example, the real [[smooth function]]s with a [[compact support]] (that is, they are zero outside some [[compact set]]) form a function space that is at the basis of the theory of [[distribution (mathematics)|distributions]].

Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and [[topology|topological]] properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of [[ordinary differential equation|ordinary]] or [[partial differential equation]]s result of the study of function spaces.

==Multi-valued functions==
{{main|Multi-valued function}}
[[File:Function with two values 1.svg|thumb|right|Together, the two square roots of all nonnegative real numbers form a single smooth curve.]]
[[File:Xto3minus3x.svg|thumb|right]]
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a [[neighbourhood (mathematics)|neighbourhood]] of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point <math>x_0,</math> there are several possible starting values for the function.

For example, in defining the [[square root]] as the inverse function of the square function, for any positive real number <math>x_0,</math> there are two choices for the value of the square root, one of which is positive and denoted <math>\sqrt {x_0},</math> and another which is negative and denoted <math>-\sqrt {x_0}.</math> These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single [[smooth curve]]. It is therefore often useful to consider these two square root functions as a single function that has two values for positive {{mvar|x}}, one value for 0 and no value for negative {{mvar|x}}.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the [[implicit function]] that maps {{mvar|y}} to a [[root of a function|root]] {{mvar|x}} of <math>x^3-3x-y =0</math> (see the figure on the right). For {{math|1=''y'' = 0}} one may choose either <math>0, \sqrt 3,\text{ or } -\sqrt 3</math> for {{mvar|x.}} By the [[implicit function theorem]], each choice defines a function; for the first one, the (maximal) domain is the interval {{math|[–2, 2]}} and the image is {{math|[–1, 1]}}; for the second one, the domain is {{math|[–2, ∞)}} and the image is {{math|[1, ∞)}}; for the last one, the domain is {{math|(–∞, 2]}} and the image is {{math|(–∞, –1]}}. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single ''multi-valued function'' of {{mvar|y}} that has three values for {{math|–2 < ''y'' < 2}}, and only one value for {{math|''y'' ≤ –2}} and {{math|''y'' ≥ –2}}.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically [[analytic function]]s. The domain to which a complex function may be extended by [[analytic continuation]] generally consists of almost the whole [[complex plane]]. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets {{mvar|i}} for the square root of –1; while, when extending through complex numbers with negative imaginary parts, one gets {{math|–''i''}}. There are generally two ways of solving the problem. One may define a function that is not [[continuous function|continuous]] along some curve, called a [[branch cut]]. Such a function is called the [[principal value]] of the function. The other way is to consider that one has a ''multi-valued function'', which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the [[monodromy]].

==In the foundations of mathematics and set theory==

The definition of a function that is given in this article requires the concept of [[set (mathematics)|set]], since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example, the [[singleton set]] may be considered as a function <math>x\mapsto \{x\}.</math> Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.<ref>{{harvnb |Gödel |1940 |p=16}}; {{harvnb |Jech |2003 |p=11}}; {{harvnb |Cunningham |2016 |p=57}}</ref>

These generalized functions may be critical in the development of a formalization of the [[foundations of mathematics]]. For example, [[Von Neumann–Bernays–Gödel set theory]], is an extension of the set theory in which the collection of all sets is a [[Class (set theory)|class]]. This theory includes the [[Von Neumann–Bernays–Gödel set theory#NBG's axiom of replacement|replacement axiom]], which may be stated as: If {{mvar|X}} is a set and {{mvar|F}} is a function, then {{math|''F''[''X'']}} is a set.

== In computer science ==
{{main|Function (programming)|Lambda calculus}}
In [[computer programming]], a [[Function (programming)|function]] is, in general, a piece of a [[computer program]], which [[implementation|implements]] the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many [[programming language]]s every [[subroutine]] is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the [[computer memory]].

[[Functional programming]] is the [[programming paradigm]] consisting of building programs by using only subroutines that behave like mathematical functions. For example, <code>if_then_else</code> is a function that takes three functions as arguments, and, depending on the result of the first function (''true'' or ''false''), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier [[program proof]]s, as being based on a well founded theory, the [[lambda calculus]] (see below).

Except for computer-language terminology, "function" has the usual mathematical meaning in [[computer science]]. In this area, a property of major interest is the [[computable function|computability]] of a function. For giving a precise meaning to this concept, and to the related concept of [[algorithm]], several [[models of computation]] have been introduced, the old ones being [[μ-recursive function|general recursive function]]s, [[lambda calculus]] and [[Turing machine]]. The fundamental theorem of [[computability theory]] is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The [[Church–Turing thesis]] is the claim that every philosophically acceptable definition of a ''computable function'' defines also the same functions.

General recursive functions are [[partial function]]s from integers to integers that can be defined from 
* [[constant function]]s, 
* [[successor function|successor]], and
* [[projection function|projection]] functions 
via the operators
* [[#Function composition|composition]], 
* [[primitive recursion]], and 
* [[μ operator|minimization]]. 
Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:
* a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...),
* every sequence of symbols may be coded as a sequence of [[bit]]s,
* a bit sequence can be interpreted as the [[binary representation]] of an integer.

[[Lambda calculus]] is a theory that defines computable functions without using [[set theory]], and is the theoretical background of functional programming. It consists of ''terms'' that are either variables, function definitions (''{{lambda}}''-terms), or applications of functions to terms. Terms are manipulated through some rules, (the {{math|''α''}}-equivalence, the {{mvar|β}}-reduction, and the {{mvar|η}}-conversion), which are the [[axiom]]s of the theory and may be interpreted as rules of computation.

In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of ''type'' in [[typed lambda calculus]]. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.

==See also==
===Subpages===
{{div col|colwidth=22em}}
* [[List of types of functions]]
* [[List of functions]]
* [[Function fitting]]
* [[Implicit function]]
{{div col end}}

===Generalizations===
{{div col|colwidth=22em}}
* [[Homomorphism]]
* [[Morphism]]
* [[Microfunction]]
* [[Distribution (mathematics)|Distribution]]
* [[Functor]]
{{div col end}}

===Related topics===
{{div col|colwidth=22em}}
* [[Associative array]]
* [[Functional (mathematics)|Functional]]
* [[Functional decomposition]]
* [[Functional predicate]]
* [[Functional programming]]
* [[Parametric equation]]
* [[Elementary function]]
* [[Closed-form expression]]<!-- a function may or may not be defined bye closed form -->
{{div col end}}

==Notes==
{{Reflist|group=note}}
==References==
{{Reflist}}

==Sources==
* {{cite book |ref=harv |last=Bartle |first=Robert |authorlink=Robert G. Bartle |title=The Elements of Real Analysis |year=1967|publisher=John Wiley & Sons}}
* {{cite book |ref=harv |last=Bloch |first=Ethan D. |title=Proofs and Fundamentals: A First Course in Abstract Mathematics |publisher=Springer |year=2011|isbn=978-1-4419-7126-5 |url=https://books.google.com/books?id=QJ_537n8zKYC}}
* {{cite book |ref=harv |last=Cunningham |first=Daniel W. |title=Set theory: A First Course |publisher=Cambridge University Press |year=2016 |isbn=978-1-107-12032-7}}
* {{cite book |ref=harv |last=Gödel |first=Kurt | authorlink=Kurt Gödel|title=The Consistency of the Continuum Hypothesis |publisher=Princeton University Press |year=1940 |isbn=978-0-691-07927-1}}
* {{cite book |ref=harv |last=Halmos |first=Paul R. |authorlink=Paul Halmos |year=1970 |title=Naive Set Theory |publisher=Springer-Verlag |isbn=978-0-387-90092-6 |url=https://books.google.com/books?id=x6cZBQ9qtgoC}}
* {{cite book |ref=harv |last=Jech | first=Thomas| authorlink=Thomas Jech| title=Set theory| edition=Third Millennium| publisher=[[Springer-Verlag]]| year=2003| isbn=978-3-540-44085-7}}
* {{cite book |ref=harv |title=Calculus |first=Michael |last=Spivak |authorlink=Michael Spivak |edition=4th |year=2008 |publisher=Publish or Perish |isbn=978-0-914098-91-1 |url=https://books.google.com/books?id=7JKVu_9InRUC}}

==Further reading==
* {{Cite book |last=Anton |first=Howard |title=Calculus with Analytical Geometry |year=1980 |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-03248-9}}
* {{Cite book |last=Bartle |first=Robert G. |title=The Elements of Real Analysis |edition=2nd |year=1976 |publisher=Wiley |isbn=978-0-471-05464-1}}
* {{Cite book |title=The Concept of Function: Aspects of Epistemology and Pedagogy|publisher=Mathematical Association of America |year=1992 |first1=Ed |last1=Dubinsky |first2=Guershon |last2=Harel |isbn=978-0-88385-081-7}}
* {{Cite book |last=Hammack |first=Richard |title=Book of Proof |year=2009 |publisher=[[Virginia Commonwealth University]] |url=http://www.people.vcu.edu/~rhammack/BookOfProof/ |chapter=12. Functions |chapter-url=http://www.people.vcu.edu/~rhammack/BookOfProof/Functions.pdf |accessdate=2012-08-01}}
* {{Cite book |last=Husch |first=Lawrence S. |title=Visual Calculus |year=2001 |publisher=[[University of Tennessee]] |url=http://archives.math.utk.edu/visual.calculus/ |accessdate=2007-09-27}}
* {{Cite book |last=Katz |first=Robert |title=Axiomatic Analysis |year=1964 |publisher=[[D. C. Heath and Company]]}}
* {{Cite journal |title=Evolution of the Function Concept: A Brief Survey |first=Israel |last=Kleiner |journal=The College Mathematics Journal |volume=20 |issue=4 |year=1989 |pages=282–300 |doi=10.2307/2686848|jstor=2686848 |citeseerx=10.1.1.113.6352 }}
* {{Cite book |title=The Cambridge History of Science: The modern physical and mathematical sciences |chapter=Between rigor and applications: Developments in the concept of function in mathematical analysis |first=Jesper |last=Lützen |chapter-url=https://books.google.com/books?id=B3WvWhJTTX8C&pg=PA468 |editor-first=Roy|editor-last=Porter |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-57199-9}} An approachable and diverting historical presentation.
* {{Cite journal |title=Historical and pedagogical aspects of the definition of function |last=Malik |first=M. A. |journal=International Journal of Mathematical Education in Science and Technology |volume=11 |issue=4 |year=1980 |pages=489–492 |doi=10.1080/0020739800110404}}
* Reichenbach, Hans (1947) ''Elements of Symbolic Logic'', Dover Publishing Inc., New York, {{isbn|0-486-24004-5}}.
* {{Cite journal |last=Ruthing |first=D. |title=Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N. |journal=Mathematical Intelligencer |volume=6 |issue=4 |pages=72–77 |year=1984}}
* {{Cite book |last1=Thomas |first1=George B. |last2=Finney |first2=Ross L. |title=Calculus and Analytic Geometry |edition=9th |year=1995 |publisher=[[Addison-Wesley]] |isbn=978-0-201-53174-9}}

==External links==
{{Commonscat|Functions (mathematics)}}
* {{springer|title=Function|id=p/f041940}}
* {{MathWorld| title=Function| id=Function}}
* [http://functions.wolfram.com/ The Wolfram Functions Site] gives formulae and visualizations of many mathematical functions.
* [https://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
{{Authority control}}

{{DEFAULTSORT:Function (Mathematics)}}
[[Category:Functions and mappings| ]]
[[Category:Basic concepts in set theory]]
[[Category:Elementary mathematics]]