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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'
Codomain. ''Encyclopedia of Mathematics''
/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time.
Historically History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well ...
, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, and this greatly enlarged the domains of application of the concept. A function is most often denoted by letters such as , and , and the value of a function at an element of its domain is denoted by ; the numerical value resulting from the ''function evaluation'' at a particular input value is denoted by replacing with this value; for example, the value of at is denoted by . When the function is not named and is represented by an expression , the value of the function at, say, may be denoted by . For example, the value at of the function that maps to (x+1)^2 may be denoted by \left.(x+1)^2\right\vert_ (which results in A function is uniquely represented by the set of all pairs , called the '' graph of the function'', a popular means of illustrating the function.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). 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 is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.


Definition

A function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to a set is an assignment of an element of to each element of . The set is called the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of the function and the set is called the codomain of the function. A function, its domain, and its codomain, are declared by the notation , and the value of a function at an element of , denoted by , is called the ''image'' of under , or the ''value'' of applied to the ''argument'' . Functions are also called '' maps'' or ''mappings'', though some authors make some distinction between "maps" and "functions" (see ). Two functions and are equal if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, given and , we have if and only if for all .This follows from the axiom of extensionality, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example, The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set. Typically, this occurs in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, where "a function often refers to a function that may have a proper subsetcalled the ''domain of definition'' by some authors, notably computer science of as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable. However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, but only that the domain is a set of real numbers that contains a non-empty open interval. Such a function is then called a partial function. For example, if is a function that has the real numbers as domain and codomain, then a function mapping the value to the value is a function from the reals to the reals, whose domain is the set of the reals , such that . The range or image of a function is the set of the images of all elements in the domain.


Total, univalent relation

Any subset of the Cartesian product of two sets and defines a binary relation between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function given above. A binary relation is univalent (also called right-unique) if :\forall x\in X, \forall y\in Y, \forall z\in Y, \quad ((x,y)\in R \land (x,z)\in R)\implies y=z. A binary relation is total if :\forall x\in X, \exists y\in Y, \quad(x,y)\in R. A partial function is a binary relation that is univalent, and a function is a binary relation that is univalent and total. Various properties of functions and function composition may be reformulated in the language of relations. Gunther Schmidt( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
For example, a function is injective if the converse relation is univalent, where the converse relation is defined as


Set exponentiation

The set of all functions from a set X to a set Y is commonly denoted as :Y^X, which is read as Y ''to the power'' X. This notation is the same as the notation for the Cartesian product of a family of copies of Y indexed by X: :Y^X=\prod_Y. The identity of these two notations is motivated by the fact that a function f can be identified with the element of the Cartesian product such that the component of index x is f(x). When Y has two elements, Y^X is commonly denoted 2^X and called the powerset of . It can be identified with the set of all
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of X, through the one-to-one correspondence that associates to each subset S\subseteq X the function f such that f(x)=1 if x\in S and f(x)=0 otherwise.


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

In functional notation, the function is immediately given a name, such as , and its definition is given by what does to the explicit argument , using a formula in terms of . For example, the function which takes a real number as input and outputs that number plus 1 is denoted by :f(x)=x+1. If a function is defined in this notation, its domain and codomain are implicitly taken to both be \R, the set of real numbers. If the formula cannot be evaluated at all real numbers, then the domain is implicitly taken to be the maximal subset of \R on which the formula can be evaluated; see Domain of a function. A more complicated example is the function :f(x)=\sin(x^2+1). In this example, the function takes a real number as input, squares it, then adds 1 to the result, then takes the sine of the result, and returns the final result as the output. 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 instead of . Functional notation was first used by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
in 1734. 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 "" for the sine function, in contrast to italic font for single-letter symbols. When using this notation, one often encounters the abuse of notation whereby the notation can refer to the value of at , or to the function itself. If the variable was previously declared, then the notation unambiguously means the value of at . Otherwise, it is useful to understand the notation as being both simultaneously; this allows one to denote composition of two functions and in a succinct manner by the notation . However, distinguishing and can become important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a ''
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
''.) Other approaches of notating functions, detailed below, avoid this problem but are less commonly used.


Arrow notation

Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, 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 of \R is implied. The domain and codomain can also be explicitly stated, for example: :\begin \operatorname\colon \Z &\to \Z\\ x &\mapsto x^2.\end This defines a function from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f\colon X\times X\to Y;\;(x,t) \mapsto f(x,t) is a function in two variables, and we want to refer to a partially applied function X\to Y produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted x\mapsto f(x,t_0) using the arrow notation. The expression x\mapsto f(x,t_0) (read: "the map taking to ") represents this new function with just one argument, whereas the expression refers to the value of the function at the


Index notation

Index notation is often used instead of functional notation. That is, instead of writing , one writes f_x. This is typically the case for functions whose domain is the set of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s. Such a function is called a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
, and, in this case the element f_n is called the th element of the sequence. The index notation is also often used for distinguishing some variables called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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 x\mapsto f(x,t) (see above) would be denoted f_t using index notation, if we define the collection of maps f_t by the formula f_t(x)=f(x,t) for all x,t\in X.


Dot notation

In the notation x\mapsto f(x), the symbol does not represent any value, it is simply a
placeholder Placeholder may refer to: Language * Placeholder name, a term or terms referring to something or somebody whose name is not known or, in that particular context, is not significant or relevant. * Filler text, text generated to fill space or provi ...
meaning that, if 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, may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function from its value at . For example, a(\cdot)^2 may stand for the function x\mapsto ax^2, and \int_a^ f(u)\,du may stand for a function defined by an integral with variable upper bound: x\mapsto \int_a^x f(u)\,du.


Specialized notations

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and
application Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a c ...
. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.


Other terms

A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular ''map'' is often used in place of ''homomorphism'' for the sake of succinctness (e.g., linear map or ''map from to '' instead of '' group homomorphism from to ''). Some authors reserve the word ''mapping'' for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term ''mapping'' for more general functions. In the theory of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map. Whichever definition of ''map'' is used, related terms like ''
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
'', '' codomain'', '' injective'', ''
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
'' have the same meaning as for a function.


Specifying a function

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


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 A = \, then one can define a function f\colon A \to \mathbb by f(1) = 2, f(2) = 3, f(3) = 4.


By a formula

Functions are often defined by a formula 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, f can be defined by the formula f(n) = n+1, for n\in\. 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 zeros of auxiliary functions. Similarly, if
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s occur in the definition of a function from \mathbb to \mathbb, 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)=\sqrt defines a function f\colon \mathbb \to \mathbb whose domain is \mathbb, because 1+x^2 is always positive if is a real number. On the other hand, f(x)=\sqrt defines a function from the reals to the reals whose domain is reduced to the interval . (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 is a function that may be written f(x) = ax^2+bx+c, where are
constants Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
. *More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
to nonnegative integers. For example, f(x) = x^3-3x-1, and f(x) = (x-1)(x^3+1) +2x^2 -1. *A rational function is the same, with divisions also allowed, such as f(x) = \frac, and f(x) = \frac 1+\frac 3x-\frac 2. *An algebraic function is the same, with th roots and roots of polynomials also allowed. *An elementary functionHere "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. is the same, with
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s and exponential functions allowed.


Inverse and implicit functions

A function f\colon X\to Y, with domain and codomain , is bijective, if for every in , there is one and only one element in such that . In this case, the inverse function of is the function f^\colon Y \to X that maps y\in Y to the element x\in X such that . For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, that maps the real numbers onto the positive numbers. If a function f\colon X\to Y is not bijective, it may occur that one can select subsets E\subseteq X and F\subseteq Y such that the restriction of to is a bijection from to , and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called
arccosine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ...
, maps onto . The other inverse trigonometric functions are defined similarly. More generally, given a binary relation between two sets and , let be a subset of such that, for every x\in E, there is some y\in Y such that . If one has a criterion allowing selecting such an for every x\in E, this defines a function f\colon E\to Y, called an implicit function, because it is implicitly defined by the relation . For example, the equation of the unit circle x^2+y^2=1 defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and . In this example, the equation can be solved in , giving y=\pm \sqrt, but, in more complicated examples, this is impossible. For example, the relation y^5+y+x=0 defines as an implicit function of , called the
Bring radical In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of the polynomial : x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thu ...
, which has \mathbb R as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and th roots. The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.


Using differential calculus

Many functions can be defined as the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of another function. This is the case of the natural logarithm, which is the antiderivative of that is 0 for . Another common example is the error function. More generally, many functions, including most special functions, can be defined as solutions of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. The simplest example is probably the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .
Power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
can be used to define functions on the domain in which they converge. For example, the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
is given by e^x = \sum_^ . 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the disc of convergence of the series. Then
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
, the exponential and the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
of a complex number.


By recurrence

Functions whose domain are the nonnegative integers, known as
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s, are often defined by recurrence relations. The factorial function on the nonnegative integers (n\mapsto n!) is a basic example, as it can be defined by the recurrence relation :n!=n(n-1)!\quad\text\quad n>0, and the initial condition :0!=1.


Representing a function

A 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 charts.


Graphs and plots

Given a function f\colon X\to Y, its ''graph'' is, formally, the set :G=\. In the frequent case where and are subsets of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (or may be identified with such subsets, e.g. intervals), an element (x,y)\in G may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a 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 :x\mapsto x^2, consisting of all points with coordinates (x, x^2) for x\in \R, yields, when depicted in Cartesian coordinates, the well known
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. If the same quadratic function x\mapsto x^2, with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates (r,\theta) =(x,x^2), the plot obtained is
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...
.


Tables

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 f\colon\^2 \to \mathbb defined as f(x,y)=xy can be represented by the familiar multiplication table 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: 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

Bar charts are often used for representing functions whose domain is a finite set, the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, or the integers. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function, , is represented by a rectangle whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -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 , there is a unique function, called the , or empty map, from the empty set to . The graph of an empty function is the empty set.By definition, the graph of the empty function to is a subset of the Cartesian product , and this product is empty. 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 ordered triplet (or equivalent ones), there is exactly one empty function for each set, thus the empty function \varnothing \mapsto X is not equal to \varnothing \mapsto Y if and only if X\ne Y, although their graph are both the empty set. * For every set and every singleton set , there is a unique function from to , which maps every element of to . This is a surjection (see below) unless is the empty set. * Given a function f\colon X\to Y, the ''canonical surjection'' of onto its image f(X)=\ is the function from to that maps to . * For every
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of a set , the inclusion map of into is the injective (see below) function that maps every element of to itself. * The identity function on a set , often denoted by , is the inclusion of into itself.


Function composition

Given two functions f\colon X\to Y and g\colon Y\to Z such that the domain of is the codomain of , their ''composition'' is the function g \circ f\colon X \rightarrow Z defined by :(g \circ f)(x) = g(f(x)). That is, the value of g \circ f is obtained by first applying to to obtain and then applying to the result to obtain . In the notation the function that is applied first is always written on the right. The composition g\circ f is an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both g \circ f and f \circ g satisfy these conditions, the composition is not necessarily commutative, that is, the functions g \circ f and f \circ g need not be equal, but may deliver different values for the same argument. For example, let and , then g(f(x))=x^2+1 and f(g(x)) = (x+1)^2 agree just for x=0. The function composition is associative in the sense that, if one of (h\circ g)\circ f and h\circ (g\circ f) is defined, then the other is also defined, and they are equal. Thus, one writes :h\circ g\circ f = (h\circ g)\circ f = h\circ (g\circ f). The identity functions \operatorname_X and \operatorname_Y are respectively a right identity and a
left identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
for functions from to . That is, if is a function with domain , and codomain , one has f\circ \operatorname_X = \operatorname_Y \circ f = f. 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, .


Image and preimage

Let f\colon X\to Y. The ''image'' under of an element of the domain is . If is any subset of , then the ''image'' of under , denoted , is the subset of the codomain consisting of all images of elements of , that is, :f(A)=\. The ''image'' of is the image of the whole domain, that is, . It is also called the range of , although the term ''range'' may also refer to the codomain.''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) On the other hand, the '' inverse image'' or '' preimage'' under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by f^(y) and is given by the equation :f^(y) = \. Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to . It is denoted by f^(B) and is given by the equation :f^(B) = \. For example, the preimage of \ under the square function is the set \. By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage f^(y) of an element of the codomain may be
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then f^(0) = \mathbb. If f\colon X\to Y is a function, and are subsets of , and and are subsets of , then one has the following properties: * A\subseteq B \Longrightarrow f(A)\subseteq f(B) * C\subseteq D \Longrightarrow f^(C)\subseteq f^(D) * A \subseteq f^(f(A)) * C \supseteq f(f^(C)) * f(f^(f(A)))=f(A) * f^(f(f^(C)))=f^(C) The preimage by of an element of the codomain is sometimes called, in some contexts, the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
of under . If a function has an inverse (see below), this inverse is denoted f^. In this case f^(C) may denote either the image by f^ or the preimage by of . This is not a problem, as these sets are equal. The notation f(A) and f^(C) may be ambiguous in the case of sets that contain some subsets as elements, such as \. In this case, some care may be needed, for example, by using square brackets f f^ /math> for images and preimages of subsets and ordinary parentheses for images and preimages of elements.


Injective, surjective and bijective functions

Let f\colon X\to Y be a function. The function is '' injective'' (or ''one-to-one'', or is an ''injection'') if for any two different elements and of . Equivalently, is injective if and only if, for any y\in Y, the preimage f^(y) contains at most one element. An empty function is always injective. If is not the empty set, then is injective if and only if there exists a function g\colon Y\to X such that g\circ f=\operatorname_X, that is, if has a left inverse. ''Proof'': If is injective, for defining , one chooses an element x_0 in (which exists as is supposed to be nonempty),The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is not needed here, as the choice is done in a single set.
and one defines by g(y)=x if y=f(x) and g(y)=x_0 if y\not\in f(X). Conversely, if g\circ f=\operatorname_X, and y=f(x), then x=g(y), and thus f^(y)=\. The function is '' surjective'' (or ''onto'', or is a ''surjection'') if its range f(X) equals its codomain Y, that is, if, for each element y of the codomain, there exists some element x of the domain such that f(x) = y (in other words, the preimage f^(y) of every y\in Y is nonempty). If, as usual in modern mathematics, the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is assumed, then is surjective if and only if there exists a function g\colon Y\to X such that f\circ g=\operatorname_Y, that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by g(y)=x, where x is an ''arbitrarily chosen'' element of f^(y). The function is '' bijective'' (or is a ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective. That is, is bijective if, for any y\in Y, the preimage f^(y) contains exactly one element. The function is bijective if and only if it admits an inverse function, that is, a function g\colon Y\to X such that g\circ f=\operatorname_X and f\circ g=\operatorname_Y. (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward). Every function f\colon X\to Y may be factorized as the composition i\circ s of a surjection followed by an injection, where is the canonical surjection of onto and is the canonical injection of into . This is the ''canonical factorization'' of . "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 Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook ...
and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps ''onto'' " differs from " maps ''into'' ", in that the former implies that is surjective, while the latter makes no assertion about the nature of . 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

If f\colon X \to Y is a function and ''S'' is a subset of ''X'', then the ''restriction'' of f to ''S'', denoted f, _S, is the function from ''S'' to ''Y'' defined by :f, _S(x) = f(x) for all ''x'' in ''S''. Restrictions can be used to define partial inverse functions: if there is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''S'' of the domain of a function f such that f, _S is injective, then the canonical surjection of f, _S onto its image f, _S(S) = f(S) is a bijection, and thus has an inverse function from f(S) to ''S''. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called
arccosine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ...
and is denoted . Function restriction may also be used for "gluing" functions together. Let X=\bigcup_U_i be the decomposition of as a union of subsets, and suppose that a function f_i\colon U_i \to Y is defined on each U_i such that for each pair i, j of indices, the restrictions of f_i and f_j to U_i \cap U_j are equal. Then this defines a unique function f\colon X \to Y such that f, _ = f_i for all . This is the way that functions on manifolds are defined. An ''extension'' of a function is a function such that is a restriction of . A typical use of this concept is the process of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane. Here is another classical example of a function extension that is encountered when studying homographies of the real line. A ''homography'' is a function h(x)=\frac such that . Its domain is the set of all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s different from -d/c, and its image is the set of all real numbers different from a/c. If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting h(\infty)=a/c and h(-d/c)=\infty.


Multivariate function

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. More formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function. The Cartesian product X_1\times\cdots\times X_n of sets X_1, \ldots, X_n is the set of all -tuples (x_1, \ldots, x_n) such that x_i\in X_i for every with 1 \leq i \leq n. Therefore, a function of variables is a function :f\colon U\to Y, where the domain has the form :U\subseteq X_1\times\cdots\times X_n. When using function notation, one usually omits the parentheses surrounding tuples, writing f(x_1,x_2) instead of f((x_1,x_2)). In the case where all the X_i are equal to the set \R of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, one has a function of several real variables. If the X_i are equal to the set \C of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, one has a
function of several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variab ...
. It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair of integers with to a pair of integers called the ''quotient'' and the ''remainder'': :\begin \text\colon\quad \Z\times (\Z\setminus \) &\to \Z\times\Z\\ (a,b) &\mapsto (\operatorname(a,b),\operatorname(a,b)). \end The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, or more generally a manifold, a vector-valued function is often called a vector field.


In calculus

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. 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, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, when the word ''function'' is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.


Real function

A ''real function'' is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s that contains an 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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
, differentiable, and even
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
. 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 and are functions, their sum, difference and product are functions defined by :\begin (f+g)(x)&=f(x)+g(x)\\ (f-g)(x)&=f(x)-g(x)\\ (f\cdot g)(x)&=f(x)\cdot g(x)\\ \end. The domains of the resulting functions are the intersection of the domains of and . The quotient of two functions is defined similarly by :\frac fg(x)=\frac, but the domain of the resulting function is obtained by removing the zeros of from the intersection of the domains of and . The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function x\mapsto \frac 1x, whose graph is a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
, and whose domain is the whole real line except for 0. The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a real differentiable function is a real function. An
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of a continuous real function is a real function that has the original function as a derivative. For example, the function x\mapsto\frac 1x is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for , is a differentiable function called the natural logarithm. A real function is
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
in an interval if the sign of \frac does not depend of the choice of and 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 is monotonic in an interval , it has an inverse function, which is a real function with domain and image . This is how inverse trigonometric functions are defined in terms of
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
. Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. For example, the sine and the cosine functions are the solutions of the linear differential equation :y''+y=0 such that :\sin 0=0, \quad \cos 0=1, \quad\frac(0)=1, \quad\frac(0)=0.


Vector-valued function

When the elements of the codomain of a function are 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 \mathbb^n or other spaces that share geometric or topological properties of \mathbb^n, such as manifolds. These vector-valued functions are given the name ''vector fields''.


Function space

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, and more specifically in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a 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 distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.


Multi-valued functions

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 A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point x_0, there are several possible starting values for the function. For example, in defining the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
as the inverse function of the square function, for any positive real number x_0, there are two choices for the value of the square root, one of which is positive and denoted \sqrt , and another which is negative and denoted -\sqrt . 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 , one value for 0 and no value for negative . In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps to a root of x^3-3x-y =0 (see the figure on the right). For one may choose either 0, \sqrt 3,\text -\sqrt 3 for . By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is . 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 that has three values for , and only one value for and . Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s. The domain to which a complex function may be extended by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...
generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets . There are generally two ways of solving the problem. One may define a function that is not
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a ''multi-valued function'', which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.


In the foundations of mathematics and set theory

The definition of a function that is given in this article requires the concept of
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, 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 x\mapsto \. 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.; ; These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
. This theory includes the
replacement axiom In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
, which may be stated as: If is a set and is a function, then is a set.


In computer science

In
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, a function is, in general, a piece of a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. Computer programs are one component of software, which also includes software documentation, documentation and oth ...
, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every
subroutine In computer programming, a function or subroutine is a sequence of program instructions that performs a specific task, packaged as a unit. This unit can then be used in programs wherever that particular task should be performed. Functions may ...
is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions tha ...
is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else 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 (see below). Except for computer-language terminology, "function" has the usual mathematical meaning in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a ''computable function'' defines also the same functions. General recursive functions are partial functions from integers to integers that can be defined from * constant functions, * successor, and * projection functions via the operators * composition, * primitive recursion, and * 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 bits, * 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 is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, and is the theoretical background of functional programming. It consists of ''terms'' that are either variables, function definitions ('-terms), or applications of functions to terms. Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation. In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of ''type'' in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.


See also


Subpages

*
List of types of functions Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function. Relative to set theory These properties concern the domai ...
* List of functions * Function fitting * Implicit function


Generalizations

* Higher-order function *
Homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
* Morphism * Microfunction *
Distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
* Functor


Related topics

* Associative array *
Closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
* Elementary function *
Functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
* Functional decomposition * Functional predicate *
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions tha ...
* Parametric equation * Set function * Simple function


Notes


References


Sources

* * * * * * *


Further reading

* * * * * * * * An approachable and diverting historical presentation. * * Reichenbach, Hans (1947) ''Elements of Symbolic Logic'', Dover Publishing Inc., New York, . * *


External links

*
The Wolfram Functions Site
gives formulae and visualizations of many mathematical functions.
NIST Digital Library of Mathematical Functions
{{Authority control Basic concepts in set theory Elementary mathematics