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Function (mathematics)

From Wikipedia, the free encyclopedia
(Redirected fromMultivariate function)
Association of one output to each input
"f(x)" redirects here. For the musical group, seef(x) (group).
Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
 Classes/properties 
  Constructions
  Generalizations  
  List of specific functions

Inmathematics, afunction from asetX to a setY assigns to each element ofX exactly one element ofY.[1] The setX is called thedomain of the function[2] and the setY is called thecodomain of the function.[3]

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of aplanet is afunction of time.Historically, the concept was elaborated with theinfinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered weredifferentiable (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 ofset theory, and this greatly increased the possible applications of the concept.

A function is often denoted by a letter such asf,g orh. The value of a functionf at an elementx of its domain (that is, the element of the codomain that is associated withx) is denoted byf(x); for example, the value off atx = 4 is denoted byf(4). Commonly, a specific function is defined by means of anexpression depending onx, such asf(x)=x2+1;{\displaystyle f(x)=x^{2}+1;} in this case, some computation, calledfunction evaluation, may be needed for deducing the value of the function at a particular value; for example, iff(x)=x2+1,{\displaystyle f(x)=x^{2}+1,} thenf(4)=42+1=17.{\displaystyle f(4)=4^{2}+1=17.}

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

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

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

Definition

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Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output
The red curve is thegraph of a function, because anyvertical line has exactly one crossing point with the curve.

Afunctionf from asetX to a setY is an assignment of one element ofY to each element ofX. The setX is called thedomain of the function and the setY is called thecodomain of the function.

If the elementy inY is assigned tox inX by the functionf, one says thatfmapsx toy, and this is commonly writteny=f(x).{\displaystyle y=f(x).} In this notation,x is theargument orvariable of the function.

A specific elementx ofX is avalue of the variable, and the corresponding element ofY is thevalue of the function atx, or theimage ofx under the function. Theimage of a function, sometimes called itsrange, is the set of the images of all elements in the domain.[6][7][8][9]

A functionf, its domainX, and its codomainY are often specified by the notationf:X→Y.{\displaystyle f:X\to Y.} One may writex↦y{\displaystyle x\mapsto y} instead ofy=f(x){\displaystyle y=f(x)}, where the symbol↦{\displaystyle \mapsto } (read 'maps to') is used to specify where a particular elementx in the domain is mapped to byf. This allows the definition of a function without naming. For example, thesquare function is the functionx↦x2.{\displaystyle x\mapsto x^{2}.}

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, iff:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } is areal function, the determination of the domain of the functionx↦1/f(x){\displaystyle x\mapsto 1/f(x)} requires knowing thezeros off. This is one of the reasons for which, inmathematical analysis, "a functionfromX toY " may refer to a function having a proper subset ofX as a domain.[note 2] For example, a "function from the reals to the reals" may refer to areal-valued function of areal variable whose domain is a proper subset of thereal numbers, typically a subset that contains a non-emptyopen interval. Such a function is then called apartial function.

A functionf on a setS means a function from the domainS, without specifying a codomain. However, some authors use it as shorthand for saying that the function isf :S →S.

Formal definition

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Diagram of a function
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.

The above definition of a function is essentially that of the founders ofcalculus,Leibniz,Newton andEuler. However, it cannot beformalized, 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 ofset theory. This set-theoretic definition is based on the fact that a function establishes arelation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, abinary relation between two setsX andY is asubset of the set of allordered pairs(x,y){\displaystyle (x,y)} such thatx∈X{\displaystyle x\in X} andy∈Y.{\displaystyle y\in Y.} The set of all these pairs is called theCartesian product ofX andY and denotedX×Y.{\displaystyle X\times Y.} Thus, the above definition may be formalized as follows.

Afunction with domainX and codomainY is a binary relationR betweenX andY that satisfies the two following conditions:[10]

This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (includingset-builder notation):

A function is formed by three sets, thedomainX,{\displaystyle X,} thecodomainY,{\displaystyle Y,} and thegraphR{\displaystyle R} that satisfy the three following conditions.

Partial functions

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Main article:Partial function

Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, apartial function fromX toY is a binary relationR betweenX andY such that, for everyx∈X,{\displaystyle x\in X,} there isat most oney inY such that(x,y)∈R.{\displaystyle (x,y)\in R.}

Using functional notation, this means that, givenx∈X,{\displaystyle x\in X,} eitherf(x){\displaystyle f(x)} is inY, or it is undefined.

The set of the elements ofX such thatf(x){\displaystyle f(x)} is defined and belongs toY is called thedomain of definition of the function. A partial function fromX toY is thus a ordinary function that has as its domain a subset ofX called the domain of definition of the function. If the domain of definition equalsX, one often says that the partial function is atotal function.

In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

Incalculus, areal-valued function of a real variable orreal function is a partial function from the setR{\displaystyle \mathbb {R} } of thereal numbers to itself. Given a real functionf:x↦f(x){\displaystyle f:x\mapsto f(x)} itsmultiplicative inversex↦1/f(x){\displaystyle x\mapsto 1/f(x)} is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute thezeros of the function, the values where the function is defined but not its multiplicative inverse.

Similarly, afunction of a complex variable is generally a partial function with a domain of definition included in the setC{\displaystyle \mathbb {C} } of thecomplex numbers. The difficulty of determining the domain of definition of acomplex function is illustrated by the multiplicative inverse of theRiemann zeta function: the determination of the domain of definition of the functionz↦1/ζ(z){\displaystyle z\mapsto 1/\zeta (z)} is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, theRiemann hypothesis.

Incomputability theory, ageneral recursive function is a partial function from the integers to the integers whose values can be computed by analgorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether0 belongs to its domain of definition (seeHalting problem).

Multivariate functions

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Not to be confused withMultivalued function.
A binary operation is a typical example of a bivariate function which assigns to each pair(x,y){\displaystyle (x,y)} the resultx∘y{\displaystyle x\circ y}.

Amultivariate function,multivariable function, orfunction 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 ofn variables is a function whose domain is a set ofn-tuples.[note 3] For example, multiplication ofintegers is a function of two variables, orbivariate function, whose domain is the set of allordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for everybinary operation. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining aparametric surface, as used in, e.g.,bivariate interpolation.

Commonly, ann-tuple is denoted enclosed between parentheses, such as in(1,2,…,n).{\displaystyle (1,2,\ldots ,n).} When usingfunctional notation, one usually omits the parentheses surrounding tuples, writingf(x1,…,xn){\displaystyle f(x_{1},\ldots ,x_{n})} instead off((x1,…,xn)).{\displaystyle f((x_{1},\ldots ,x_{n})).}

Givenn setsX1,…,Xn,{\displaystyle X_{1},\ldots ,X_{n},} the set of alln-tuples(x1,…,xn){\displaystyle (x_{1},\ldots ,x_{n})} such thatx1∈X1,…,xn∈Xn{\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}} is called theCartesian product ofX1,…,Xn,{\displaystyle X_{1},\ldots ,X_{n},} and denotedX1×⋯×Xn.{\displaystyle X_{1}\times \cdots \times X_{n}.}

Therefore, a multivariate function is a function that has a Cartesian product or aproper subset of a Cartesian product as a domain.

f:U→Y,{\displaystyle f:U\to Y,}

where the domainU has the form

U⊆X1×⋯×Xn.{\displaystyle U\subseteq X_{1}\times \cdots \times X_{n}.}

If all theXi{\displaystyle X_{i}} are equal to the setR{\displaystyle \mathbb {R} } of thereal numbers or to the setC{\displaystyle \mathbb {C} } of thecomplex numbers, one talks respectively of afunction of several real variables or of afunction of several complex variables.

Notation

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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

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The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letterf. 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

f(x),sin⁡(3),orf(x2+1).{\displaystyle f(x),\quad \sin(3),\quad {\text{or}}\quad f(x^{2}+1).}

The argument between the parentheses may be avariable, oftenx, that represents an arbitrary element of the domain of the function, a specific element of the domain (3 in the above example), or anexpression that can be evaluated to an element of the domain (x2+1{\displaystyle x^{2}+1} in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "letf(x)=sin⁡(x2+1){\displaystyle f(x)=\sin(x^{2}+1)}".

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 writesinx instead ofsin(x).

Functional notation was first used byLeonhard Euler in 1734.[11] 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, aroman type is customarily used instead, such as "sin" for thesine function, in contrast to italic font for single-letter symbols.

The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "letf(x){\displaystyle f(x)} be a function". This is anabuse of notation that is useful for a simpler formulation.

Arrow notation

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Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "maps to". For example,x↦x+1{\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain ofR{\displaystyle \mathbb {R} } is implied.

The domain and codomain can also be explicitly stated, for example:

sqr:Z→Zx↦x2.{\displaystyle {\begin{aligned}\operatorname {sqr} \colon \mathbb {Z} &\to \mathbb {Z} \\x&\mapsto x^{2}.\end{aligned}}}

This defines a functionsqr from the integers to the integers that returns the square of its input.

As a common application of the arrow notation, supposef:X×X→Y;(x,t)↦f(x,t){\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)} is a function in two variables, and we want to refer to apartially applied functionX→Y{\displaystyle X\to Y} produced by fixing the second argument to the valuet0 without introducing a new function name. The map in question could be denotedx↦f(x,t0){\displaystyle x\mapsto f(x,t_{0})} using the arrow notation. The expressionx↦f(x,t0){\displaystyle x\mapsto f(x,t_{0})} (read: "the map takingx tof ofx commat nought") represents this new function with just one argument, whereas the expressionf(x0,t0) refers to the value of the functionf at thepoint(x0,t0).

Index notation

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Index notation may be used instead of functional notation. That is, instead of writingf (x), one writesfx.{\displaystyle f_{x}.}

This is typically the case for functions whose domain is the set of thenatural numbers. Such a function is called asequence, and, in this case the elementfn{\displaystyle f_{n}} is called thenth element of the sequence.

The index notation can also be used for distinguishing some variables calledparameters 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 mapx↦f(x,t){\displaystyle x\mapsto f(x,t)} (see above) would be denotedft{\displaystyle f_{t}} using index notation, if we define the collection of mapsft{\displaystyle f_{t}} by the formulaft(x)=f(x,t){\displaystyle f_{t}(x)=f(x,t)} for allx,t∈X{\displaystyle x,t\in X}.

Dot notation

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In the notationx↦f(x),{\displaystyle x\mapsto f(x),}the symbolx does not represent any value; it is simply aplaceholder, meaning that, ifx 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,x may be replaced by any symbol, often aninterpunct " ⋅". This may be useful for distinguishing the functionf (⋅) from its valuef (x) atx.

For example,a(⋅)2{\displaystyle a(\cdot )^{2}} may stand for the functionx↦ax2{\displaystyle x\mapsto ax^{2}}, and∫a(⋅)f(u)du{\textstyle \int _{a}^{\,(\cdot )}f(u)\,du} may stand for a function defined by anintegral with variable upper bound:x↦∫axf(u)du{\textstyle x\mapsto \int _{a}^{x}f(u)\,du}.

Specialized notations

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There are other, specialized notations for functions in sub-disciplines of mathematics. For example, inlinear algebra andfunctional analysis,linear forms and thevectors they act upon are denoted using adual pair to show the underlyingduality. This is similar to the use ofbra–ket notation in quantum mechanics. Inlogic and thetheory of computation, the function notation oflambda calculus is used to explicitly express the basic notions of functionabstraction andapplication. Incategory theory andhomological algebra, networks of functions are described in terms of how they and their compositionscommute with each other usingcommutative diagrams that extend and generalize the arrow notation for functions described above.

Functions of more than one variable

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In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a functionf can be defined as mapping any pair of real numbers(x,y){\displaystyle (x,y)} to the sum of their squares,x2+y2{\displaystyle x^{2}+y^{2}}. Such a function is commonly written asf(x,y)=x2+y2{\displaystyle f(x,y)=x^{2}+y^{2}} and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such asf(w,x,y){\displaystyle f(w,x,y)},f(w,x,y,z){\displaystyle f(w,x,y,z)}.

Other terms

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For broader coverage of this topic, seeMap (mathematics).
TermDistinction from "function"
Map/MappingNone; the terms are synonymous.[12]
A map can haveany set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set ofreal orcomplex numbers.[13]
Alternatively, a map is associated with aspecial structure (e.g. by explicitly specifying a structured codomain in its definition). For example, alinear map.[14]
HomomorphismA function between twostructures of the same type that preserves the operations of the structure (e.g. agroup homomorphism).[15]
MorphismA generalisation of homomorphisms to anycategory, even when the objects of the category are not sets (for example, agroup defines a category with only one object, which has the elements of the group as morphisms; seeCategory (mathematics) § Examples for this example and other similar ones).[16]

A function may also be called amap or amapping, 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 particularmap may be used in place ofhomomorphism for the sake of succinctness (e.g.,linear map ormap fromG toH instead ofgroup homomorphism fromG toH). Some authors[14] reserve the wordmapping for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such asSerge Lang,[13] use "function" only to refer to maps for which thecodomain is a subset of thereal orcomplex numbers, and use the termmapping for more general functions.

In the theory ofdynamical systems, a map denotes anevolution function used to creatediscrete dynamical systems. See alsoPoincaré map.

Whichever definition ofmap is used, related terms likedomain,codomain,injective,continuous have the same meaning as for a function.

Specifying a function

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Given a functionf{\displaystyle f}, by definition, to each elementx{\displaystyle x} of the domain of the functionf{\displaystyle f}, there is a unique element associated to it, the valuef(x){\displaystyle f(x)} off{\displaystyle f} atx{\displaystyle x}. There are several ways to specify or describe howx{\displaystyle x} is related tof(x){\displaystyle f(x)}, both explicitly and implicitly. Sometimes, a theorem or anaxiom 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 functionf{\displaystyle f}.

By listing function values

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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, ifA={1,2,3}{\displaystyle A=\{1,2,3\}}, then one can define a functionf:A→R{\displaystyle f:A\to \mathbb {R} } byf(1)=2,f(2)=3,f(3)=4.{\displaystyle f(1)=2,f(2)=3,f(3)=4.}

By a formula

[edit]

Functions are often defined by anexpression that describes a combination ofarithmetic 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,f{\displaystyle f} can be defined by the formulaf(n)=n+1{\displaystyle f(n)=n+1}, forn∈{1,2,3}{\displaystyle n\in \{1,2,3\}}.

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 thezeros of auxiliary functions. Similarly, ifsquare roots occur in the definition of a function fromR{\displaystyle \mathbb {R} } toR,{\displaystyle \mathbb {R} ,} 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,f(x)=1+x2{\displaystyle f(x)={\sqrt {1+x^{2}}}} defines a functionf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } whose domain isR,{\displaystyle \mathbb {R} ,} because1+x2{\displaystyle 1+x^{2}} is always positive ifx is a real number. On the other hand,f(x)=1−x2{\displaystyle f(x)={\sqrt {1-x^{2}}}} defines a function from the reals to the reals whose domain is reduced to the interval[−1, 1]. (In old texts, such a domain was called thedomain of definition of the function.)

Functions can be classified by the nature of formulas that define them:

Inverse and implicit functions

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A functionf:X→Y,{\displaystyle f:X\to Y,} with domainX and codomainY, isbijective, if for everyy inY, there is one and only one elementx inX such thaty =f(x). In this case, theinverse function off is the functionf−1:Y→X{\displaystyle f^{-1}:Y\to X} that mapsy∈Y{\displaystyle y\in Y} to the elementx∈X{\displaystyle x\in X} such thaty =f(x). For example, thenatural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called theexponential function, that maps the real numbers onto the positive numbers.

If a functionf:X→Y{\displaystyle f:X\to Y} is not bijective, it may occur that one can select subsetsE⊆X{\displaystyle E\subseteq X} andF⊆Y{\displaystyle F\subseteq Y} such that therestriction off toE is a bijection fromE toF, and has thus an inverse. Theinverse trigonometric functions are defined this way. For example, thecosine function induces, by restriction, a bijection from theinterval[0,π] onto the interval[−1, 1], and its inverse function, calledarccosine, maps[−1, 1] onto[0,π]. The other inverse trigonometric functions are defined similarly.

More generally, given abinary relationR between two setsX andY, letE be a subset ofX such that, for everyx∈E,{\displaystyle x\in E,} there is somey∈Y{\displaystyle y\in Y} such thatx R y. If one has a criterion allowing selecting such ay for everyx∈E,{\displaystyle x\in E,} this defines a functionf:E→Y,{\displaystyle f:E\to Y,} called animplicit function, because it is implicitly defined by the relationR.

For example, the equation of theunit circlex2+y2=1{\displaystyle x^{2}+y^{2}=1} defines a relation on real numbers. If−1 <x < 1 there are two possible values ofy, one positive and one negative. Forx = ± 1, these two values become both equal to 0. Otherwise, there is no possible value ofy. This means that the equation defines two implicit functions with domain[−1, 1] and respective codomains[0, +∞) and(−∞, 0].

In this example, the equation can be solved iny, givingy=±1−x2,{\displaystyle y=\pm {\sqrt {1-x^{2}}},} but, in more complicated examples, this is impossible. For example, the relationy5+y+x=0{\displaystyle y^{5}+y+x=0} definesy as an implicit function ofx, called theBring radical, which hasR{\displaystyle \mathbb {R} } as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations andnth roots.

Theimplicit function theorem provides milddifferentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

Using differential calculus

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Many functions can be defined as theantiderivative of another function. This is the case of thenatural logarithm, which is the antiderivative of1/x that is 0 forx = 1. Another common example is theerror function.

More generally, many functions, including mostspecial functions, can be defined as solutions ofdifferential equations. The simplest example is probably theexponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 forx = 0.

Power series can be used to define functions on the domain in which they converge. For example, theexponential function is given byex=∑n=0∞xnn!{\textstyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}. 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 itsTaylor series in some interval, this power series allows immediately enlarging the domain to a subset of thecomplex numbers, thedisc of convergence of the series. Thenanalytic continuation allows enlarging further the domain for including almost the wholecomplex plane. This process is the method that is generally used for defining thelogarithm, theexponential and thetrigonometric functions of a complex number.

By recurrence

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Main article:Recurrence relation

Functions whose domain are the nonnegative integers, known assequences, are sometimes defined byrecurrence relations.

Thefactorial function on the nonnegative integers (n↦n!{\displaystyle n\mapsto n!}) is a basic example, as it can be defined by the recurrence relation

n!=n(n−1)!forn>0,{\displaystyle n!=n(n-1)!\quad {\text{for}}\quad n>0,}

and the initial condition

0!=1.{\displaystyle 0!=1.}

Representing a function

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Agraph 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 bybar charts.

Graphs and plots

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Main article:Graph of a function
The function mapping each year to its US motor vehicle death count, shown as aline chart
The same function, shown as a bar chart

Given a functionf:X→Y,{\displaystyle f:X\to Y,} itsgraph is, formally, the set

G={(x,f(x))∣x∈X}.{\displaystyle G=\{(x,f(x))\mid x\in X\}.}

In the frequent case whereX andY are subsets of thereal numbers (or may be identified with such subsets, e.g.intervals), an element(x,y)∈G{\displaystyle (x,y)\in G} may be identified with a point having coordinatesx,y in a 2-dimensional coordinate system, e.g. theCartesian plane. Parts of this may create aplot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called thegraph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of thesquare function

x↦x2,{\displaystyle x\mapsto x^{2},}

consisting of all points with coordinates(x,x2){\displaystyle (x,x^{2})} forx∈R,{\displaystyle x\in \mathbb {R} ,} yields, when depicted in Cartesian coordinates, the well knownparabola. If the same quadratic functionx↦x2,{\displaystyle x\mapsto x^{2},} with the same formal graph, consisting of pairs of numbers, is plotted instead inpolar coordinates(r,θ)=(x,x2),{\displaystyle (r,\theta )=(x,x^{2}),} the plot obtained isFermat's spiral.

Tables

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Main article: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 functionf:{1,…,5}2→R{\displaystyle f:\{1,\ldots ,5\}^{2}\to \mathbb {R} } defined asf(x,y)=xy{\displaystyle f(x,y)=xy} can be represented by the familiarmultiplication table

y
x
12345
112345
2246810
33691215
448121620
5510152025

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:

xsinx
1.2890.960557
1.2900.960835
1.2910.961112
1.2920.961387
1.2930.961662

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

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Main article:Bar chart

A bar chart can represent a function whose domain is a finite set, thenatural numbers, or theintegers. In this case, an elementx of the domain is represented by aninterval of thex-axis, and the corresponding value of the function,f(x), is represented by arectangle whose base is the interval corresponding tox and whose height isf(x) (possibly negative, in which case the bar extends below thex-axis).

General properties

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

Standard functions

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There are a number of standard functions that occur frequently:

  • For every setX, there is a unique function, called theempty function, orempty map, from theempty set toX. The graph of an empty function is the empty set.[note 5] 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 anordered triplet (or equivalent ones), there is exactly one empty function for each set, thus the empty function∅→X{\displaystyle \varnothing \to X} is not equal to∅→Y{\displaystyle \varnothing \to Y} if and only ifX≠Y{\displaystyle X\neq Y}, although their graphs are both theempty set.
  • For every setX and everysingleton set{s}, there is a unique function fromX to{s}, which maps every element ofX tos. This is a surjection (see below) unlessX is the empty set.
  • Given a functionf:X→Y,{\displaystyle f:X\to Y,} thecanonical surjection off onto its imagef(X)={f(x)∣x∈X}{\displaystyle f(X)=\{f(x)\mid x\in X\}} is the function fromX tof(X) that mapsx tof(x).
  • For everysubsetA of a setX, theinclusion map ofA intoX is the injective (see below) function that maps every element ofA to itself.
  • Theidentity function on a setX, often denoted byidX, is the inclusion ofX into itself.

Function composition

[edit]
Main article:Function composition

Given two functionsf:X→Y{\displaystyle f:X\to Y} andg:Y→Z{\displaystyle g:Y\to Z} such that the domain ofg is the codomain off, theircomposition is the functiong∘f:X→Z{\displaystyle g\circ f:X\rightarrow Z} defined by

(g∘f)(x)=g(f(x)).{\displaystyle (g\circ f)(x)=g(f(x)).}

That is, the value ofg∘f{\displaystyle g\circ f} is obtained by first applyingf tox to obtainy =f(x) and then applyingg to the resulty to obtaing(y) =g(f(x)). In this notation, the function that is applied first is always written on the right.

The compositiong∘f{\displaystyle g\circ f} is anoperation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when bothg∘f{\displaystyle g\circ f} andf∘g{\displaystyle f\circ g} satisfy these conditions, the composition is not necessarilycommutative, that is, the functionsg∘f{\displaystyle g\circ f} andf∘g{\displaystyle f\circ g} need not be equal, but may deliver different values for the same argument. For example, letf(x) =x2 andg(x) =x + 1, theng(f(x))=x2+1{\displaystyle g(f(x))=x^{2}+1} andf(g(x))=(x+1)2{\displaystyle f(g(x))=(x+1)^{2}} agree just forx=0.{\displaystyle x=0.}

The function composition isassociative in the sense that, if one of(h∘g)∘f{\displaystyle (h\circ g)\circ f} andh∘(g∘f){\displaystyle h\circ (g\circ f)} is defined, then the other is also defined, and they are equal, that is,(h∘g)∘f=h∘(g∘f).{\displaystyle (h\circ g)\circ f=h\circ (g\circ f).} Therefore, it is usual to just writeh∘g∘f.{\displaystyle h\circ g\circ f.}

Theidentity functionsidX{\displaystyle \operatorname {id} _{X}} andidY{\displaystyle \operatorname {id} _{Y}} are respectively aright identity and aleft identity for functions fromX toY. That is, iff is a function with domainX, and codomainY, one hasf∘idX=idY∘f=f.{\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}

  • A composite function g(f(x)) can be visualized as the combination of two "machines".
    A composite functiong(f(x)) can be visualized as the combination of two "machines".
  • A simple example of a function composition
    A simple example of a function composition
  • Another composition. In this example, (g ∘ f )(c) = #.
    Another composition. In this example,(g ∘ f )(c) = #.

Image and preimage

[edit]
Main article:Image (mathematics)

Letf:X→Y.{\displaystyle f:X\to Y.} Theimage underf of an elementx of the domainX isf(x).[6] IfA is any subset ofX, then theimage ofA underf, denotedf(A), is the subset of the codomainY consisting of all images of elements ofA,[6] that is,

f(A)={f(x)∣x∈A}.{\displaystyle f(A)=\{f(x)\mid x\in A\}.}

Theimage off is the image of the whole domain, that is,f(X).[17] It is also called therange off,[6][7][8][9] although the termrange may also refer to the codomain.[9][17][18]

On the other hand, theinverse image orpreimage underf of an elementy of the codomainY is the set of all elements of the domainX whose images underf equaly.[6] In symbols, the preimage ofy is denoted byf−1(y){\displaystyle f^{-1}(y)} and is given by the equation

f−1(y)={x∈X∣f(x)=y}.{\displaystyle f^{-1}(y)=\{x\in X\mid f(x)=y\}.}

Likewise, the preimage of a subsetB of the codomainY is the set of the preimages of the elements ofB, that is, it is the subset of the domainX consisting of all elements ofX whose images belong toB.[6] It is denoted byf−1(B){\displaystyle f^{-1}(B)} and is given by the equation

f−1(B)={x∈X∣f(x)∈B}.{\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}

For example, the preimage of{4,9}{\displaystyle \{4,9\}} under thesquare function is the set{−3,−2,2,3}{\displaystyle \{-3,-2,2,3\}}.

By definition of a function, the image of an elementx of the domain is always a single element of the codomain. However, the preimagef−1(y){\displaystyle f^{-1}(y)} of an elementy of the codomain may beempty or contain any number of elements. For example, iff is the function from the integers to themselves that maps every integer to 0, thenf−1(0)=Z{\displaystyle f^{-1}(0)=\mathbb {Z} }.

Iff:X→Y{\displaystyle f:X\to Y} is a function,A andB are subsets ofX, andC andD are subsets ofY, then one has the following properties:

The preimage byf of an elementy of the codomain is sometimes called, in some contexts, thefiber ofy underf.

If a functionf has an inverse (see below), this inverse is denotedf−1.{\displaystyle f^{-1}.} In this casef−1(C){\displaystyle f^{-1}(C)} may denote either the image byf−1{\displaystyle f^{-1}} or the preimage byf ofC. This is not a problem, as these sets are equal. The notationf(A){\displaystyle f(A)} andf−1(C){\displaystyle f^{-1}(C)} may be ambiguous in the case of sets that contain some subsets as elements, such as{x,{x}}.{\displaystyle \{x,\{x\}\}.} In this case, some care may be needed, for example, by using square bracketsf[A],f−1[C]{\displaystyle f[A],f^{-1}[C]} for images and preimages of subsets and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions

[edit]
Main article:Bijection, injection and surjection

Letf:X→Y{\displaystyle f:X\to Y} be a function.

The functionf isinjective (orone-to-one, or is aninjection) iff(a) ≠f(b) for every two different elementsa andb ofX.[17][19] Equivalently,f is injective if and only if, for everyy∈Y,{\displaystyle y\in Y,} the preimagef−1(y){\displaystyle f^{-1}(y)} contains at most one element. An empty function is always injective. IfX is not the empty set, thenf is injective if and only if there exists a functiong:Y→X{\displaystyle g:Y\to X} such thatg∘f=idX,{\displaystyle g\circ f=\operatorname {id} _{X},} that is, iff has aleft inverse.[19]Proof: Iff is injective, for definingg, one chooses an elementx0{\displaystyle x_{0}} inX (which exists asX is supposed to be nonempty),[note 6] and one definesg byg(y)=x{\displaystyle g(y)=x} ify=f(x){\displaystyle y=f(x)} andg(y)=x0{\displaystyle g(y)=x_{0}} ify∉f(X).{\displaystyle y\not \in f(X).} Conversely, ifg∘f=idX,{\displaystyle g\circ f=\operatorname {id} _{X},} andy=f(x),{\displaystyle y=f(x),} thenx=g(y),{\displaystyle x=g(y),} and thusf−1(y)={x}.{\displaystyle f^{-1}(y)=\{x\}.}

The functionf issurjective (oronto, or is asurjection) if its rangef(X){\displaystyle f(X)} equals its codomainY{\displaystyle Y}, that is, if, for each elementy{\displaystyle y} of the codomain, there exists some elementx{\displaystyle x} of the domain such thatf(x)=y{\displaystyle f(x)=y} (in other words, the preimagef−1(y){\displaystyle f^{-1}(y)} of everyy∈Y{\displaystyle y\in Y} is nonempty).[17][20] If, as usual in modern mathematics, theaxiom of choice is assumed, thenf is surjective if and only if there exists a functiong:Y→X{\displaystyle g:Y\to X} such thatf∘g=idY,{\displaystyle f\circ g=\operatorname {id} _{Y},} that is, iff has aright inverse.[20] The axiom of choice is needed, because, iff is surjective, one definesg byg(y)=x,{\displaystyle g(y)=x,} wherex{\displaystyle x} is anarbitrarily chosen element off−1(y).{\displaystyle f^{-1}(y).}

The functionf isbijective (or is abijection or aone-to-one correspondence) if it is both injective and surjective.[17][21] That is,f is bijective if, for everyy∈Y,{\displaystyle y\in Y,} the preimagef−1(y){\displaystyle f^{-1}(y)} contains exactly one element. The functionf is bijective if and only if it admits aninverse function, that is, a functiong:Y→X{\displaystyle g:Y\to X} such thatg∘f=idX{\displaystyle g\circ f=\operatorname {id} _{X}} andf∘g=idY.{\displaystyle f\circ g=\operatorname {id} _{Y}.}[21] (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).

Every functionf:X→Y{\displaystyle f:X\to Y} may befactorized as the compositioni∘s{\displaystyle i\circ s} of a surjection followed by an injection, wheres is the canonical surjection ofX ontof(X) andi is the canonical injection off(X) intoY. This is thecanonical factorization off.

"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 theBourbaki group and imported into English.[22] 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 "f mapsXontoY" differs from "f mapsXintoB", in that the former implies thatf is surjective, while the latter makes no assertion about the nature off. 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

[edit]
Main article:Restriction (mathematics)

Iff:X→Y{\displaystyle f:X\to Y} is a function andS is a subset ofX, then therestriction off{\displaystyle f} toS, denotedf|S{\displaystyle f|_{S}}, is the function fromS toY defined by

f|S(x)=f(x){\displaystyle f|_{S}(x)=f(x)}

for allx inS. Restrictions can be used to define partialinverse functions: if there is asubsetS of the domain of a functionf{\displaystyle f} such thatf|S{\displaystyle f|_{S}} is injective, then the canonical surjection off|S{\displaystyle f|_{S}} onto its imagef|S(S)=f(S){\displaystyle f|_{S}(S)=f(S)} is a bijection, and thus has an inverse function fromf(S){\displaystyle f(S)} toS. One application is the definition ofinverse trigonometric functions. For example, thecosine function is injective when restricted to theinterval[0,π]. The image of this restriction is the interval[−1, 1], and thus the restriction has an inverse function from[−1, 1] to[0,π], which is calledarccosine and is denotedarccos.

Function restriction may also be used for "gluing" functions together. LetX=⋃i∈IUi{\textstyle X=\bigcup _{i\in I}U_{i}} be the decomposition ofX as aunion of subsets, and suppose that a functionfi:Ui→Y{\displaystyle f_{i}:U_{i}\to Y} is defined on eachUi{\displaystyle U_{i}} such that for each pairi,j{\displaystyle i,j} of indices, the restrictions offi{\displaystyle f_{i}} andfj{\displaystyle f_{j}} toUi∩Uj{\displaystyle U_{i}\cap U_{j}} are equal. Then this defines a unique functionf:X→Y{\displaystyle f:X\to Y} such thatf|Ui=fi{\displaystyle f|_{U_{i}}=f_{i}} for alli. This is the way that functions onmanifolds are defined.

Anextension of a functionf is a functiong such thatf is a restriction ofg. A typical use of this concept is the process ofanalytic continuation, that allows extending functions whose domain is a small part of thecomplex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studyinghomographies of thereal line. Ahomography is a functionh(x)=ax+bcx+d{\displaystyle h(x)={\frac {ax+b}{cx+d}}} such thatad −bc ≠ 0. Its domain is the set of allreal numbers different from−d/c,{\displaystyle -d/c,} and its image is the set of all real numbers different froma/c.{\displaystyle a/c.} If one extends the real line to theprojectively extended real line by including∞, one may extendh to a bijection from the extended real line to itself by settingh(∞)=a/c{\displaystyle h(\infty )=a/c} andh(−d/c)=∞{\displaystyle h(-d/c)=\infty }.

In calculus

[edit]
Further information:History of the function concept

The idea of function, starting in the 17th century, was fundamental to the newinfinitesimal calculus. At that time, onlyreal-valued functions of areal variable were considered, and all functions were assumed to besmooth. But the definition was soon extended tofunctions of several variables and tofunctions 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 introductorycalculus, when the wordfunction 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 withSTEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such asreal analysis andcomplex analysis.

Real function

[edit]
See also:Real analysis
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions:sine andcosine.

Areal function is areal-valuedfunction of a real variable, that is, a function whose codomain is thefield of real numbers and whose domain is a set ofreal numbers that contains aninterval. In this section, these functions are simply calledfunctions.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they arecontinuous,differentiable, and evenanalytic. This regularity insures that these functions can be visualized by theirgraphs. In this section, all functions are differentiable in some interval.

Functions enjoypointwise operations, that is, iff andg are functions, their sum, difference and product are functions defined by

(f+g)(x)=f(x)+g(x)(f−g)(x)=f(x)−g(x)(f⋅g)(x)=f(x)⋅g(x).{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}

The domains of the resulting functions are theintersection of the domains off andg. The quotient of two functions is defined similarly by

fg(x)=f(x)g(x),{\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}

but the domain of the resulting function is obtained by removing thezeros ofg from the intersection of the domains off andg.

Thepolynomial functions are defined bypolynomials, and their domain is the whole set of real numbers. They includeconstant functions,linear functions andquadratic functions.Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoiddivision by zero. The simplest rational function is the functionx↦1x,{\displaystyle x\mapsto {\frac {1}{x}},} whose graph is ahyperbola, and whose domain is the wholereal line except for 0.

Thederivative of a real differentiable function is a real function. Anantiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the functionx↦1x{\textstyle x\mapsto {\frac {1}{x}}} is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero forx = 1, is a differentiable function called thenatural logarithm.

A real functionf ismonotonic in an interval if the sign off(x)−f(y)x−y{\displaystyle {\frac {f(x)-f(y)}{x-y}}} does not depend of the choice ofx andy 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 functionf is monotonic in an intervalI, it has aninverse function, which is a real function with domainf(I) and imageI. This is howinverse trigonometric functions are defined in terms oftrigonometric 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 abijection between the real numbers and the positive real numbers. This inverse is theexponential function.

Many other real functions are defined either by theimplicit function theorem (the inverse function is a particular instance) or as solutions ofdifferential equations. For example, thesine and thecosine functions are the solutions of thelinear differential equation

y″+y=0{\displaystyle y''+y=0}

such that

sin⁡0=0,cos⁡0=1,∂sin⁡x∂x(0)=1,∂cos⁡x∂x(0)=0.{\displaystyle \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.}

Vector-valued function

[edit]
Main articles:Vector-valued function andVector field

When the elements of the codomain of a function arevectors, 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 itsvelocity vector is a vector-valued function.

Some vector-valued functions are defined on a subset ofRn{\displaystyle \mathbb {R} ^{n}} or other spaces that share geometric ortopological properties ofRn{\displaystyle \mathbb {R} ^{n}}, such asmanifolds. These vector-valued functions are given the namevector fields.

Function space

[edit]
Main articles:Function space andFunctional analysis

Inmathematical analysis, and more specifically infunctional analysis, afunction space is a set ofscalar-valued orvector-valued functions, which share a specific property and form atopological vector space. For example, the realsmooth functions with acompact support (that is, they are zero outside somecompact set) form a function space that is at the basis of the theory ofdistributions.

Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic andtopological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions ofordinary orpartial differential equations result of the study of function spaces.

Multi-valued functions

[edit]
Main article:Multi-valued function
Together, the two square roots of all nonnegative real numbers form a single smooth curve.

Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on aneighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting pointx0,{\displaystyle x_{0},} there are several possible starting values for the function.

For example, in defining thesquare root as the inverse function of the square function, for any positive real numberx0,{\displaystyle x_{0},} there are two choices for the value of the square root, one of which is positive and denotedx0,{\displaystyle {\sqrt {x_{0}}},} and another which is negative and denoted−x0.{\displaystyle -{\sqrt {x_{0}}}.} 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 singlesmooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positivex, one value for 0 and no value for negativex.

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 theimplicit function that mapsy to arootx ofx3−3x−y=0{\displaystyle x^{3}-3x-y=0} (see the figure on the right). Fory = 0 one may choose either0,3, or −3{\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}} forx. By theimplicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval[−2, 2] and the image is[−1, 1]; for the second one, the domain is[−2, ∞) and the image is[1, ∞); for the last one, the domain is(−∞, 2] and the image is(−∞, −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 singlemulti-valued function ofy that has three values for−2 <y < 2, and only one value fory ≤ −2 andy ≥ −2.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typicallyanalytic functions. The domain to which a complex function may be extended byanalytic continuation generally consists of almost the wholecomplex 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 getsi for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets−i. There are generally two ways of solving the problem. One may define a function that is notcontinuous along some curve, called abranch cut. Such a function is called theprincipal value of the function. The other way is to consider that one has amulti-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 themonodromy.

In the foundations of mathematics

[edit]

The definition of a function that is given in this article requires the concept ofset, 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, thesingleton set may be considered as a functionx↦{x}.{\displaystyle x\mapsto \{x\}.} 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 definitions for these weakly specified functions.[23]

These generalized functions may be critical in the development of a formalization of thefoundations 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 aclass. This theory includes thereplacement axiom, which may be stated as: IfX is a set andF is a function, thenF[X] is a set.

In alternative formulations of the foundations of mathematics usingtype theory rather than set theory, functions are taken asprimitive notions rather than defined from other kinds of object. They are the inhabitants offunction types, and may be constructed using expressions in thelambda calculus.[24]

In computer science

[edit]
Main articles:Function (computer programming) andLambda calculus

Incomputer programming, afunction is, in general, asubroutine whichimplements the abstract concept of function. That is, it is a program unit that produces an output for each input.Functional programming is theprogramming paradigm consisting of building programs by using only subroutines that behave like mathematical functions, meaning that they have noside effects and depend only on their arguments: they arereferentially transparent. For example,if_then_else is a function that takes three (nullary) functions as arguments, and, depending on the value of the first argument (true orfalse), returns the value of either the second or the third argument. An important advantage of functional programming is that it makes easierprogram proofs, as being based on a well founded theory, thelambda calculus (see below). However, side effects are generally necessary for practical programs, ones that performinput/output. There is a class ofpurely functional languages, such asHaskell, which encapsulate the possibility of side effects in the type of a function. Others, such as theML family, simply allow side effects.

In manyprogramming languages, every subroutine is called a function, even when there is no output but only side effects, and when the functionality consists simply of modifying some data in thecomputer memory.

Outside the context of programming languages, "function" has the usual mathematical meaning incomputer science. In this area, a property of major interest is thecomputability of a function. For giving a precise meaning to this concept, and to the related concept ofalgorithm, severalmodels of computation have been introduced, the old ones beinggeneral recursive functions,lambda calculus, andTuring machine. The fundamental theorem ofcomputability 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. TheChurch–Turing thesis is the claim that every philosophically acceptable definition of acomputable function defines also the same functions.

General recursive functions arepartial functions from integers to integers that can be defined from

via the operators

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, etc.),
  • every sequence of symbols may be coded as a sequence ofbits,
  • a bit sequence can be interpreted as thebinary representation of an integer.

Lambda calculus is a theory that defines computable functions without usingset theory, and is the theoretical background of functional programming. It consists ofterms that are either variables, function definitions (𝜆-terms), or applications of functions to terms. Terms are manipulated by interpreting itsaxioms (theα-equivalence, theβ-reduction, and theη-conversion) asrewriting rules, which can be used for 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 oftype intyped lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.

See also

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Subpages

[edit]

Generalizations

[edit]

Related topics

[edit]

Notes

[edit]
  1. ^This definition of "graph" refers to aset of pairs of objects. Graphs, in the sense ofdiagrams, 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).
  2. ^The true domain of such a function is often called thedomain of definition of the function.
  3. ^n may also be 1, thus subsuming functions as defined above. Forn = 0, eachconstant is a special case of a multivariate function, too.
  4. ^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.
  5. ^By definition, the graph of the empty function toX is a subset of the Cartesian product∅ ×X, and this product is empty.
  6. ^Theaxiom of choice is not needed here, as the choice is done in a single set.

References

[edit]
  1. ^Halmos 1970, p. 30; the wordsmap,mapping,transformation,correspondence, andoperator are sometimes used synonymously.
  2. ^Halmos 1970
  3. ^"Mapping".Encyclopedia of Mathematics.EMS Press. 2001 [1994].
  4. ^"function | Definition, Types, Examples, & Facts".Encyclopædia Britannica. Retrieved2020-08-17.
  5. ^Spivak 2008, p. 39.
  6. ^abcdefKudryavtsev, L.D. (2001) [1994]."Function".Encyclopedia of Mathematics.EMS Press.
  7. ^abTaalman, Laura; Kohn, Peter (2014).Calculus.New York City:W. H. Freeman and Company. p. 3.ISBN 978-1-4292-4186-1.LCCN 2012947365.OCLC 856545590.OL 27544563M.
  8. ^abTrench, William F. (2013) [2003].Introduction to Real Analysis (2.04th ed.).Pearson Education (originally; self-republished by the author). pp. 30–32.ISBN 0-13-045786-8.LCCN 2002032369.OCLC 953799815.Zbl 1204.00023.
  9. ^abcThomson, Brian S.; Bruckner, Judith B.;Bruckner, Andrew M. (2008) [2001].Elementary Real Analysis(PDF) (2nd ed.).Prentice Hall (originally; 2nd ed. self-republished by the authors). pp. A-4 –A-5.ISBN 978-1-4348-4367-8.OCLC 1105855173.OL 31844948M.Zbl 0872.26001.
  10. ^Halmos, Paul R. (1974).Naive Set Theory. Springer. pp. 30–33.
  11. ^Larson, Ron; Edwards, Bruce H. (2010).Calculus of a Single Variable. Cengage Learning. p. 19.ISBN 978-0-538-73552-0.
  12. ^Weisstein, Eric W."Map".Wolfram MathWorld. Retrieved2019-06-12.
  13. ^abLang, Serge (1987)."III §1. Mappings".Linear Algebra (3rd ed.). Springer. p. 43.ISBN 978-0-387-96412-6.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, orC or into a fieldK.
  14. ^abApostol, T. M. (1981).Mathematical Analysis (2nd ed.). Addison-Wesley. p. 35.ISBN 978-0-201-00288-1.OCLC 928947543.
  15. ^James, Robert C.; James, Glenn (1992).Mathematics dictionary (5th ed.). Van Nostrand Reinhold. p. 202.ISBN 0-442-00741-8.OCLC 25409557.
  16. ^James & James 1992, p. 48
  17. ^abcdeGowers, Timothy;Barrow-Green, June;Leader, Imre, eds. (2008).The Princeton Companion to Mathematics.Princeton, New Jersey:Princeton University Press. p. 11.doi:10.1515/9781400830398.ISBN 978-0-691-11880-2.JSTOR j.ctt7sd01.LCCN 2008020450.MR 2467561.OCLC 227205932.OL 19327100M.Zbl 1242.00016.
  18. ^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)
  19. ^abIvanova, O. A. (2001) [1994]."Injection".Encyclopedia of Mathematics.EMS Press.
  20. ^abIvanova, O.A. (2001) [1994]."Surjection".Encyclopedia of Mathematics.EMS Press.
  21. ^abIvanova, O.A. (2001) [1994]."Bijection".Encyclopedia of Mathematics.EMS Press.
  22. ^Hartnett, Kevin (9 November 2020)."Inside the Secret Math Society Known Simply as Nicolas Bourbaki".Quanta Magazine. Retrieved2024-06-05.
  23. ^Gödel 1940, p. 16;Jech 2003, p. 11;Cunningham 2016, p. 57
  24. ^Klev, Ansten (2019). "A comparison of type theory with set theory". In Centrone, Stefania; Kant, Deborah; Sarikaya, Deniz (eds.).Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Synthese Library. Vol. 407. Cham: Springer. pp. 271–292.doi:10.1007/978-3-030-15655-8_12.ISBN 978-3-030-15654-1.MR 4352345.

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