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{{Short description|Association of one output to each input}}
{{Functions}}

In [[Mathematics|mathematics]], a '''function''' from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} assigns to each element of {{mvar|X}} exactly one element of {{mvar|Y}}.<ref name=halmos>{{harvnb |Halmos |1970 |p=30}}; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator''{{cn|reason=Give a citation for each synonym. In particular, afaik, 'transformation', 'correspondence', and 'operator' refer to special kinds of functions, not to the general class.|date=November 2023}} are often used synonymously.</ref> The set {{mvar|X}} is called the [[Domain of a function|domain]] of the function<ref>{{harvnb|Halmos|1970}}</ref> and the set {{mvar|Y}} is called the [[Codomain|codomain]] of the function.<ref name=codomain>{{eom|title=Mapping|oldid=37940|mode=cs1}}</ref>

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a [[Astronomy:Planet|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 a function was formalized at the end of the 19th century in terms of [[Set theory|set theory]], and this greatly enlarged the domains of application of the concept.

A function is most often denoted by letters such as {{mvar|f}}, {{mvar|g}} and {{mvar|h}}, and the value of a function {{mvar|f}} at an element {{mvar|x}} of its domain is denoted by {{math|''f''(''x'')}}; the numerical value resulting from the ''function evaluation'' at a particular input value is denoted by replacing {{mvar|x}} with this value; for example, the value of {{mvar|f}} at {{math|''x'' {{=}} 4}} is denoted by {{math|''f''(4)}}. When the function is not named and is represented by an [[Expression (mathematics)|expression]] {{mvar|E}}, the value of the function at, say, {{math|''x'' {{=}} 4}} may be denoted by {{math|1={{itco|''E''}}{{!}}<sub>''x''=4</sub>}}. For example, the value at {{math|4}} of the function that maps {{mvar|x}} to <math>(x+1)^2</math> may be denoted by <math>\left.(x+1)^2\right\vert_{x=4}</math>{{cn|date=July 2022}} (which results in {{math|25).}}

A function is uniquely represented by the set of all pairs {{math|(''x'', ''f''{{hair space}}(''x''))}}, called the ''[[Graph of a function|graph of the function]]'', a popular means of illustrating 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><ref>{{Cite web|title=function {{!}} Definition, Types, Examples, & Facts| url=https://www.britannica.com/science/function-mathematics|access-date=2020-08-17|website=Encyclopedia Britannica|language=en}}</ref> 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|science]], [[Engineering:Engineering|engineering]], 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}}

== Definition ==

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

A '''function''' {{mvar|f}} from a [[Set (mathematics)|set]] {{mvar|X}} to a set {{mvar|Y}} is an assignment of an element of {{mvar|Y}} to each element of {{mvar|X}}. The set {{mvar|X}} is called the [[Domain of a function|domain]] of the function and the set {{mvar|Y}} is called the [[Codomain|codomain]] of the function.

If the element {{mvar|y}} in {{mvar|Y}} is assigned to {{mvar|x}} in {{mvar|X}} by the function {{mvar|f}}, one says that {{mvar|f}} ''maps'' {{mvar|x}} to {{mvar|y}}, and this is commonly written <math>y=f(x).</math> In this notation, {{mvar|x}} is the ''argument'' or ''variable'' of the function. A specific element {{mvar|x}} of {{mvar|X}} is a ''value of the variable'', and the corresponding element of {{mvar|Y}} is the ''value of the function'' at {{mvar|x}}, or the ''image'' of {{mvar|x}} under the function.

A function {{mvar|f}}, its domain {{mvar|X}}, and its codomain {{mvar|Y}} are often specified  by the notation <math>f: X\to Y.</math> In this case, one writes often <math>x\mapsto y</math> instead of <math>f(y).</math> This allows to define functions without naming them. For example, the square function is the function <math>x\mapsto x^2.</math>

The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if <math>f:\R\to\R</math> is a real function, the determination of the domain of the function <math>x\mapsto 1/f(x)</math> requires to know the [[Zero of a function|zeros]] of {{mvar|f.}} This is one of the reasons for which, in [[Mathematical analysis|mathematical analysis]], "a function {{nowrap|from {{mvar|X}} to {{mvar|Y}} "}} often refers to a function that may have a proper subset of {{mvar|X}} as a domain.<ref group="note">The true domain of such a function is often called the ''domain of definition'' of the function.</ref> 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]] whose domain is a proper subset of the [[Real number|real number]]s, typically a subset that contains a non-empty open interval. Such a function is then called a [[Partial function|partial function]].

The [[Range of a function|range]] or [[Image (mathematics)|image]] of a function is the set of the [[Image (mathematics)|images]] of all elements in the domain.<ref name="EOM Function"/><ref name="T&K Calc p.3">{{Taalman Kohn Calculus|p=3}}</ref><ref name="Trench RA pp.30-32">{{Trench Intro Real Analysis|pp=30–32}}</ref><ref name="TBB RA pp.A4-A5">{{Thomson Bruckner Bruckner Elementary Real Analysis|pp=A-4–A-5}}</ref>

A function {{mvar|f}} on a set {{mvar|S}} means a function from the domain {{mvar|S}}, without specifying a codomain. However, some authors use it as shorthand for saying that the function is {{math|''f'' : ''S'' → ''S''}}.

=== Formal definition ===
thumb|Diagram of a function
thumb|Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair.  Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair therein

The above definition of a function is essentially that of the founders of [[Calculus|calculus]], Leibniz, [[Biography:Isaac Newton|Newton]] and Euler. However, it cannot be [[Formal proof|formalized]], since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of [[Set theory|set theory]]. It is this set-theoretic definition that is presented here.

A [[Binary relation|binary relation]] between two sets {{math|''X''}} and {{math|''Y''}} is a [[Subset|subset]] of the set of all [[Ordered pair|ordered pair]]s <math>(x, y)</math> such that <math>x\in X</math> and <math>y\in Y.</math> The set of all these pairs is called the [[Cartesian product]] of {{math|''X''}} and {{math|''Y''}} and denoted <math>X\times Y.</math>

A ''function'' with domain {{math|''X''}} and codomain {{math|''Y''}} is a binary relation {{mvar|R}} between {{math|''X''}} and {{math|''Y''}} that satisfies the two following conditions:
* Univalent relation:<math display =block>(x,y)\in R \land (x,z)\in R \implies y=z</math>
* Total relation: <math display =block>\forall x\in X, \exists y\in Y, \left(x, y\right) \in R</math>

These conditions are exactly the formalization of the above definition of a function.

=== Multivariate functions <span class="anchor" id="MULTIVARIATE_FUNCTION"></span> ===
[[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''', '''multivariable 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 travelled and its average speed.

Formally, a function of {{mvar|n}} variables is a function whose domain is a set of {{mvar|n}}-tuples.
For example, multiplication of [[Integer|integer]]s is a function of two variables, or '''bivariate function''', whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every [[Binary operation|binary operation]]. More generally, every mathematical operation is defined as a multivariate function. Commonly, an {{mvar|n}}-tuple is denoted enclosed between parentheses, such as in <math>(1,2,\ldots, n).</math>  When using functional notation, one usually omits the parentheses surrounding tuples, writing  <math>f(x_1,\ldots,x_n)</math> instead of <math>f((x_1,\ldots,x_n)).</math>

Given {{mvar|n}} sets <math>X_1,\ldots, X_n,</math> the set of all {{mvar|n}}-tuples <math>(x_1,\ldots,x_n)</math> such that <math>x_1\in X_1, \ldots, x_n\in X_n</math> is called the [[Cartesian product]] of <math>X_1,\ldots, X_n,</math> and denoted <math>X_1\times\cdots\times X_n.</math>

Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
: <math>f: U\to Y,</math>
where the domain {{mvar|U}} has the form
: <math>U\subseteq X_1\times\cdots\times X_n.</math>

If all the <math>X_i</math> are equal to the set <math>\R</math> of the [[Real number|real number]]s or to the set <math>\C</math> of the [[Complex number|complex number]]s, one talks respectively of a [[Function of several real variables|function of several real variables]] or of a [[Function of several complex variables|function of several complex variables]].

== Notation ==

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.

=== Functional notation ===
The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter {{mvar|f}}. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in 
:<math>f(x), \quad \sin(3),\quad \text{or}\quad f(x^2+1).</math>
The argument between the parentheses may be a [[Variable (mathematics)|variable]], often {{mvar|x}}, that represents an arbitrary element of the domain of the function, a specific element of the domain ({{math|3}} in the above example), or an [[Expression (mathematics)|expression]] that can be evaluated to an element of the domain (<math>x^2+1</math> in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let <math>f(x)=\sin(x^2+1)</math>".

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write {{math|sin ''x''}} instead of {{math|sin(''x'')}}.

Functional notation was first used by [[Biography:Leonhard Euler|Leonhard Euler]] in 1734.<ref>{{cite book|first1=Ron|last1=Larson|first2=Bruce H.|last2=Edwards|title=Calculus of a Single Variable|page=19|year=2010|publisher=Cengage Learning|isbn=978-0-538-73552-0}}</ref> Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "{{math|sin}}" for the sine function, in contrast to italic font for single-letter symbols.

The functional notation is often used colloqually for referring to a function and simultaneously naming its argument, such as in "let <math>f(x)</math> be a function". This is an [[Abuse of notation|abuse of notation]] that is often useful for a simpler formulation.

=== Arrow notation ===
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, <math>x\mapsto x+1</math> is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of <math>\R</math> is implied.

The domain and codomain can also be explicitly stated, for example:
: <math>\begin{align}
\operatorname{sqr}\colon \Z &\to \Z\\
x &\mapsto x^2.\end{align}</math>

This defines a function {{math|sqr}} from the integers to the integers that returns the square of its input.

As a common application of the arrow notation, suppose <math>f: X\times X\to Y;\;(x,t) \mapsto f(x,t)</math> is a function in two variables, 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. The expression <math>x\mapsto f(x,t_0)</math> (read: "the map taking {{mvar|x}} to {{mvar|f}} of {{mvar|x}} comma {{mvar|t}} nought") represents this new function with just one argument, whereas the expression {{math|''f''(''x''<sub>0</sub>, ''t''<sub>0</sub>)}} refers to the value of the function {{mvar|f}} at the {{nowrap|point {{math|(''x''<sub>0</sub>, ''t''<sub>0</sub>)}}.}}

=== Index notation ===

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

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

The index notation is also often used for distinguishing some variables called [[Parameter|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, 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''{{hair space}}(⋅)}} from its value {{math|''f''{{hair space}}(''x'')}} at {{mvar|x}}.

For example, <math> a(\cdot)^2</math> may stand for the function <math> x\mapsto ax^2</math>, and <math display="inline"> \int_a^{\, (\cdot)} f(u)\,du</math> may stand for a function defined by an integral with variable upper bound: <math display="inline"> 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|linear algebra]] and [[Functional analysis|functional analysis]], [[Linear form|linear form]]s and the [[Vector (mathematics and physics)|vectors]] they act upon are denoted using a [[Dual pair|dual pair]] to show the underlying [[Duality (mathematics)|duality]].  This is similar to the use of [[Bra–ket notation|bra–ket notation]] in quantum mechanics.  In [[Mathematical logic|logic]] and the [[Theory of computation|theory of computation]], the function notation of [[Lambda calculus|lambda calculus]] is used to explicitly express the basic notions of function [[Abstraction (computer science)|abstraction]] and [[Function application|application]].  In [[Category theory|category theory]] and [[Homological algebra|homological algebra]], networks of functions are described in terms of how they and their compositions [[Commutative property|commute]] with each other using [[Commutative diagram|commutative diagram]]s that extend and generalize the arrow notation for functions described above.

===Functions of more than one variable===
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function {{mvar|f}} can be defined as mapping any pair of real numbers <math>(x, y)</math> to the sum of their squares, <math>x^2 + y^2</math>. Such a function is commonly written as <math>f(x, y)=x^2 + y^2</math> and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as <math>f(w,x, y)</math>, <math>f(w,x, y, z)</math>.

== 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 name=Lang87p43>{{cite book |last=Lang |first=Serge |title=Linear Algebra |chapter=III §1. Mappings |chapter-url={{GBurl|0DUXym7QWfYC|p=43}} |publisher=Springer |date=1987 |isbn=978-0-387-96412-6 |edition=3rd |page=43 |quote=A function is a special type of mapping, namely it is a mapping from a set into the set of numbers, i.e. into, '''R''', or '''C''' or into a field ''K''.}}</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|linear map]].<ref name=Apostol81p35/>
|-
|[[Homomorphism]]
|-
|[[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>{{harvnb|James|James|1992|p=48}}</ref>
|}

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|maps of manifolds]]). In particular ''map'' is often used in place of ''homomorphism'' for the sake of succinctness (e.g., [[Linear map|linear map]] or ''map from {{mvar|G}} to {{mvar|H}}'' instead of ''[[Group homomorphism|group homomorphism]] from {{mvar|G}} to {{mvar|H}}''). Some authors<ref name=Apostol81p35>{{cite book |first=T.M. |last=Apostol |title=Mathematical Analysis|year=1981 |publisher=Addison-Wesley |page=35 |isbn=978-0-201-00288-1 |oclc=928947543 |edition=2nd}}</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 [[Biography:Serge Lang|Serge Lang]],<ref name=Lang87p43/> use "function" only to refer to maps for which the [[Codomain|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|dynamical system]]s, a map denotes an 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 of a function|domain]]'', ''[[Codomain|codomain]]'', ''[[Injective function|injective]]'', ''[[Continuous function|continuous]]'' have the same meaning as for a 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|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. For example, if <math>A = \{ 1, 2, 3 \}</math>, then one can define a function <math>f: 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|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: \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 {{closed-closed|−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 define them:
* A [[Quadratic function|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|polynomial function]] is a function that can be defined by a formula involving only additions, subtractions, multiplications, and [[Exponentiation|exponentiation]] to nonnegative integer powers. For example, <math>f(x) = x^3-3x-1</math> and <math>f(x) = (x-1)(x^3+1) +2x^2 -1</math> are polynomial functions of <math>x</math>.
* A [[Rational function|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|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|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|logarithm]]s and exponential functions allowed.

=== Inverse and implicit functions ===
A function <math>f : 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|inverse function]] of {{mvar|f}} is the function <math>f^{-1} : 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|natural logarithm]] is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the [[Exponential function|exponential function]], that maps the real numbers onto the positive numbers.

If a function <math>f: 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 {{mvar|f}} to {{mvar|E}} is a bijection from {{mvar|E}} to {{mvar|F}}, and has thus an inverse. The [[Inverse trigonometric functions|inverse trigonometric functions]] are defined this way. For example, the cosine function induces, by restriction, a bijection from the [[Interval (mathematics)|interval]] {{closed-closed|0, ''π''}} onto the interval {{closed-closed|−1, 1}}, and its inverse function, called arccosine, maps {{closed-closed|−1, 1}} onto {{closed-closed|0, ''π''}}. The other inverse trigonometric functions are defined similarly.

More generally, given a [[Binary relation|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: E\to Y,</math> called an [[Implicit function|implicit function]], because it is implicitly defined by the relation {{mvar|R}}.

For example, the equation of the [[Unit circle|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 {{closed-closed|−1, 1}} and respective codomains {{closed-open|0, +∞}} and {{open-closed|−∞, 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+y+x=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|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|antiderivative]] of another function. This is the case of the [[Natural logarithm|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|error function]].

More generally, many functions, including most special functions, can be defined as solutions of [[Differential equation|differential equation]]s. The simplest example is probably the [[Exponential function|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|exponential function]] is given by <math display="inline">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|complex number]]s, the disc of convergence of the series. Then [[Analytic continuation|analytic continuation]] allows enlarging further the domain for including almost the whole [[Complex plane|complex plane]]. This process is the method that is generally used for defining the [[Logarithm|logarithm]], the [[Exponential function|exponential]] and the [[Trigonometric functions|trigonometric functions]] of a complex number.

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

The [[Factorial|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|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|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 : X\to Y,</math> its ''graph'' is, formally, the set
: <math>G=\{(x,f(x))\mid x\in X\}.</math>

In the frequent case where {{mvar|X}} and {{mvar|Y}} are subsets of the [[Real number|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|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:\{1,\ldots,5\}^2 \to \mathbb{R}</math> defined as <math>f(x,y)=xy</math> can be represented by the familiar [[Multiplication table|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|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|natural number]]s, or the [[Integer|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|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:
* For every set {{mvar|X}}, there is a unique function, called the '''{{vanchor|empty function}}''', or '''empty map''', from the [[Empty set|empty set]] to {{mvar|X}}. The graph of an empty function is the empty set.<ref group=note>By definition, the graph of the empty function to {{mvar|X}} is a subset of the Cartesian product {{math|∅ × ''X''}}, and this product is empty.</ref> The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an [[Tuple|ordered triplet]] (or equivalent ones), there is exactly one empty function for each set, thus the empty function <math>\varnothing \to X</math> is not equal to <math>\varnothing \to Y</math> if and only if <math>X\ne Y</math>, although their graphs are both the [[Empty set|empty set]].
* 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: 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|subset]] {{mvar|A}} of a set {{mvar|X}}, the [[Inclusion map|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|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: X\to Y</math> and <math>g: 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: 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, that is, <math>(h\circ g)\circ f = h\circ (g\circ f).</math> Therefore, it is usual to just write <math>h\circ g\circ f.</math>

The [[Identity function|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: X\to Y.</math> The ''image'' under {{mvar|f}} of an element {{mvar|x}} of the domain {{mvar|X}} is {{math|''f''(''x'')}}.<ref name="EOM Function"/> If {{math|''A''}} is any subset of {{math|''X''}}, then the ''image'' of {{mvar|A}} under {{mvar|f}}, denoted {{math|''f''(''A'')}}, is the subset of the codomain {{math|''Y''}} consisting of all images of elements of {{mvar|A}},<ref name="EOM Function"/> 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'')}}.{{r|PCM p.11}} It is also called the [[Range of a function|range]] of {{mvar|f}},{{r|EOM Function|T&K Calc p.3|Trench RA pp.30-32|TBB RA pp.A4-A5}} although the term ''range'' may also refer to the codomain.{{r|TBB RA pp.A4-A5|PCM p.11}}<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'' under {{mvar|f}} of an element {{mvar|y}} of the codomain {{mvar|Y}} is the set of all elements of the domain {{math|''X''}} whose images under {{mvar|f}} equal {{mvar|y}}.<ref name="EOM Function"/> In symbols, the preimage of {{mvar|y}} is denoted by <math>f^{-1}(y)</math> and is given by the equation
: <math>f^{-1}(y) = \{x \in X \mid f(x) = y\}.</math>
Likewise, the preimage of a subset {{math|''B''}} of the codomain {{math|''Y''}} is the set of the preimages of the elements of {{math|''B''}}, that is, it is the subset of the domain {{math|''X''}} consisting of all elements of {{math|''X''}} whose images belong to {{math|''B''}}.<ref name="EOM Function"/> It is denoted by <math>f^{-1}(B)</math> and is given by the equation
: <math>f^{-1}(B) = \{x \in X \mid f(x) \in B\}.</math>
For example, the preimage of <math>\{4, 9\}</math> under the square function is the set <math>\{-3,-2,2,3\}</math>.

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 <math>f^{-1}(y)</math> of an element {{mvar|y}} of the codomain 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 : 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 ===
{{main|Bijection, injection and surjection}}

Let <math>f : 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}}.<ref name="PCM p.11">{{Princeton Companion to Mathematics|p=11}}</ref><ref name="EOM Injection">{{eom |title=Injection |oldid=30986 |author-first=O.A. |author-last=Ivanova|mode=cs1}}</ref> Equivalently, {{mvar|f}} is injective if and only 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, then {{mvar|f}} is injective if and only if there exists a function <math>g: Y\to X</math> such that <math>g\circ f=\operatorname{id}_X,</math> that is, if {{mvar|f}} has a left inverse.<ref name="EOM Injection"/> ''Proof'': 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|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> Conversely, if <math>g\circ f=\operatorname{id}_X,</math> and <math>y=f(x),</math>  then <math>x=g(y),</math> and thus <math>f^{-1}(y)=\{x\}.</math>

The function {{mvar|f}} is ''surjective'' (or ''onto'', or is a ''surjection'') if its range <math>f(X)</math> equals its codomain <math>Y</math>, that is, if, for each element <math>y</math> of the codomain, there exists some element <math>x</math> of the domain such that <math>f(x) = y</math> (in other words, the preimage <math>f^{-1}(y)</math> of every <math>y\in Y</math> is nonempty).<ref name="PCM p.11"/><ref name="EOM Surjection">{{eom |title=Surjection |oldid=35689 |author-first=O.A. |author-last=Ivanova|mode=cs1}}</ref> If, as usual in modern mathematics, the [[Axiom of choice|axiom of choice]] is assumed, then {{mvar|f}} is surjective if and only if there exists a function <math>g: Y\to X</math> such that <math>f\circ g=\operatorname{id}_Y,</math> that is, if {{mvar|f}} has a right inverse.<ref name="EOM Surjection"/> 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 a ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective.<ref name="PCM p.11"/><ref name="EOM Bijection">{{eom |title=Bijection |oldid=30987 |author-first=O.A. |author-last=Ivanova|mode=cs1}}</ref> 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|inverse function]], that is, a function <math>g : Y\to X</math> such that <math>g\circ f=\operatorname{id}_X</math> and <math>f\circ g=\operatorname{id}_Y.</math><ref name="EOM Bijection"/> (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).

Every function <math>f: X\to Y</math> may be [[Factorization|factorized]] as the composition <math>i\circ s</math> 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 Bourbaki group and imported into English.{{citation needed|date=January 2021}}  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''}}. 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 of being more symmetrical.

=== Restriction and extension <span class="anchor" id="Restrictions and extensions"></span> ===
<!-- This section is linked from [[Subgroup]], Restriction, [[Quadratic form]] -->
{{main|Restriction (mathematics)}}
If <math>f : X \to Y</math> is a function and {{math|''S''}} is a subset of {{math|''X''}}, then the ''restriction'' of <math>f</math> to ''S'', denoted <math>f|_S</math>, is the function from {{math|''S''}} to {{math|''Y''}} defined by
: <math>f|_S(x) = f(x)</math>
for all {{math|''x''}} in {{math|''S''}}. Restrictions can be used to define partial [[Inverse function|inverse function]]s: if there is a [[Subset|subset]] {{math|''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 {{math|''S''}}. One application is the definition of [[Inverse trigonometric functions|inverse trigonometric functions]]. For example, the [[Cosine|cosine]] function is injective when restricted to the [[Interval (mathematics)|interval]] {{closed-closed|0, ''π''}}. The image of this restriction is the interval {{closed-closed|−1, 1}}, and thus the restriction has an inverse function from {{closed-closed|−1, 1}} to {{closed-closed|0, ''π''}}, which is called arccosine and is denoted {{math|arccos}}.

Function restriction may also be used for "gluing" functions together. Let <math display="inline"> X=\bigcup_{i\in I}U_i</math> be the decomposition of {{mvar|X}} as a union of subsets, and suppose that a function <math>f_i : 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 : 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|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|analytic continuation]], that allows extending functions whose domain is a small part of the [[Complex plane|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|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|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|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>.

== In calculus ==
{{further|History of the function concept}}

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. 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. But the definition was soon extended to 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|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|real analysis]] and [[Complex analysis|complex analysis]].

=== Real function ===
[[File:Gerade.svg|thumb|right|Graph of a linear function]]
[[File:Polynomialdeg2.svg|thumb|right|Graph of a polynomial function, here a quadratic function.]]
[[File:Sine cosine one period.svg|thumb|right|Graph of two trigonometric functions: [[Sine|sine]] and [[Cosine|cosine]].]]
A ''real function'' is a [[Real-valued function|real-valued]] [[Function of a real variable|function of a 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|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 graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, 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 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|polynomial function]]s are defined by [[Polynomial|polynomial]]s, and their domain is the whole set of real numbers. They include [[Constant function|constant function]]s, [[Linear function|linear function]]s and [[Quadratic function|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|division by zero]]. The simplest rational function is the function <math>x\mapsto \frac 1x,</math> whose graph is a [[Hyperbola|hyperbola]], and whose domain is the whole [[Real line|real line]] except for 0.

The [[Derivative|derivative]] of a real differentiable function is a real function. An [[Antiderivative|antiderivative]] of a continuous real function is a real function that has the original function as a derivative. For example, the function <math display="inline">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|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|inverse function]], which is a real function with domain {{math|''f''(''I'')}} and image {{mvar|I}}. This is how [[Inverse trigonometric functions|inverse trigonometric functions]] are defined in terms of [[Trigonometric functions|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|bijection]] between the real numbers and the positive real numbers. This inverse is the [[Exponential function|exponential function]].

Many other real functions are defined either by the [[Implicit function theorem|implicit function theorem]] (the inverse function is a particular instance) or as solutions of [[Differential equation|differential equation]]s. For example, the [[Sine|sine]] and the [[Cosine|cosine]] functions are the solutions of the [[Linear differential equation|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|mathematical analysis]], and more specifically in [[Functional analysis|functional analysis]], a '''function space''' is a set of scalar-valued or [[Vector-valued function|vector-valued function]]s, which share a specific property and form a [[Topological vector space|topological vector space]]. For example, the real smooth functions 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|partial differential equation]]s result of the study of function spaces.

== Multi-valued functions ==
[[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|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|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|implicit function theorem]], each choice defines a function; for the first one, the (maximal) domain is the interval {{closed-closed|−2, 2}} and the image is {{closed-closed|−1, 1}}; for the second one, the domain is {{closed-open|−2, ∞}} and the image is {{closed-open|1, ∞}}; for the last one, the domain is {{open-closed|−∞, 2}} and the image is {{open-closed|−∞, −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|analytic function]]s. The domain to which a complex function may be extended by [[Analytic continuation|analytic continuation]] generally consists of almost the whole [[Complex plane|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|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|monodromy]].

== In the foundations of mathematics ==

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 [[Philosophy:Foundations of mathematics|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 alternative formulations of the foundations of mathematics using [[Type theory|type theory]] rather than set theory, functions are taken as [[Primitive notion|primitive notion]]s rather than defined from other kinds of object. They are the inhabitants of [[Function type|function type]]s, and may be constructed using expressions in the [[Lambda calculus|lambda calculus]].<ref>
{{cite book
 | last = Klev | first = Ansten
 | editor1-last = Centrone | editor1-first = Stefania
 | editor2-last = Kant | editor2-first = Deborah
 | editor3-last = Sarikaya | editor3-first = Deniz
 | contribution = A comparison of type theory with set theory
 | doi = 10.1007/978-3-030-15655-8_12
 | isbn = 978-3-030-15654-1
 | location = Cham
 | mr = 4352345
 | pages = 271–292
 | publisher = Springer
 | series = Synthese Library
 | title = Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts
 | volume = 407
 | year = 2019}}</ref>

== In computer science ==
{{main|Function (computer programming)|Lambda calculus}}
In [[Computer programming|computer programming]], a function is, in general, a piece of a [[Computer program|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|programming language]]s every [[Subroutine|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|computer memory]].

[[Functional programming]] is the [[Programming paradigm|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 proofs, as being based on a well founded theory, the [[Lambda calculus|lambda calculus]] (see below).

Except for computer-language terminology, "function" has the usual mathematical meaning in [[Computer science|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|algorithm]], several models of computation have been introduced, the old ones being [[Μ-recursive function|general recursive function]]s, [[Lambda calculus|lambda calculus]] and [[Turing machine]]. The fundamental theorem of [[Computability theory|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|partial function]]s from integers to integers that can be defined from 
* [[Constant function|constant function]]s, 
* [[Successor function|successor]], and
* projection functions 
via the operators
* 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|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|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|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|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}}
* [[Higher-order function]]
* [[Homomorphism]]
* [[Morphism]]
* Microfunction
* [[Distribution (mathematics)|Distribution]]
* [[Functor]]
{{div col end}}

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

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist|refs=
<ref name="EOM Function">{{eom |title=Function |oldid=36823 |author-first=L.D. |author-last=Kudryavtsev|mode=cs1}}</ref>
}}

== Sources ==
{{refbegin}}
* {{cite book |last=Bartle |first=Robert |title=The Elements of Real Analysis |publisher=Wiley |isbn=978-0-471-05465-8 |date=1976 |edition=2nd |oclc=465115030}}
* {{cite book |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 |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 |last=Gödel |first=Kurt | title=The Consistency of the Continuum Hypothesis |publisher=Princeton University Press |year=1940 |isbn=978-0-691-07927-1}}
* {{cite book |last=Halmos |first=Paul R. |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 |last=Jech | first=Thomas|  title=Set theory| edition=3rd  |publisher=[[Physics:Springer-Verlag|Springer-Verlag]]| year=2003| isbn=978-3-540-44085-7}}
* {{cite book |title=Calculus |first=Michael |last=Spivak |edition=4th |year=2008 |publisher=Publish or Perish |isbn=978-0-914098-91-1 |url=https://books.google.com/books?id=7JKVu_9InRUC}}
{{refend}}

== Further reading ==
{{refbegin}}
* {{cite book |last=Anton |first=Howard |title=Calculus with Analytical Geometry |url=https://archive.org/details/studentssolution00anto |url-access=registration |year=1980 |publisher={{wipe|John Wiley & Sons}} |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=[[Organization:Virginia Commonwealth University|Virginia Commonwealth University]] |url=https://www.people.vcu.edu/~rhammack/BookOfProof/ |chapter=12. Functions |chapter-url=https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf#page=235 |access-date=2012-08-01}}
* {{cite book |last=Husch |first=Lawrence S. |title=Visual Calculus |year=2001 |publisher=[[Organization:University of Tennessee|University of Tennessee]] |url=http://archives.math.utk.edu/visual.calculus/ |access-date=2007-09-27}}
* {{cite book |last=Katz |first=Robert |title=Axiomatic Analysis |year=1964 |publisher=[[Software:D. C. Heath and Company|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}}
* {{cite book |last=Reichenbach |first=Hans |date=1947 |title=Elements of Symbolic Logic |publisher=Dover  |isbn=0-486-24004-5}}
* {{cite journal |last=Ruthing |first=D. |title=Old Intelligencer: Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N. |journal=Mathematical Intelligencer |volume=6 |issue=4 |pages=71–78 |year=1984 |doi=10.1007/BF03026743|s2cid=189883712 }}
* {{cite book |last1=Thomas |first1=George B. |last2=Finney |first2=Ross L. |title=Calculus and Analytic Geometry |url=https://archive.org/details/calculusanalytic00geor_0 |url-access=registration |edition=9th |year=1995 |publisher=Addison-Wesley |isbn=978-0-201-53174-9}}
{{refend}}

== External links ==
* [http://functions.wolfram.com/ The Wolfram Functions] – website giving formulae and visualizations of many mathematical functions
* [https://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]

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