Mathematical Functions

In mathematics, a function from a set 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 of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'
Codomain. ''Encyclopedia of Mathematics''
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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, 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, 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, engineering, 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 to a set is an assignment of an element of to each element of . The set is called the domain 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, 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 numbers, 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 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 subsets 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 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''.) 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 numbers. Such a function is called a sequence, 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 parameters 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 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 and functional analysis, 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 and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute 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 systems, 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'', ''codomain'', ''injective'', ''continuous'' 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 roots 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.
*More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integers. For example, $f(x) = x^3-3x-1,$ 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 logarithms 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, 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, 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, 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 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 equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .

Power series can be used to define functions on the domain in which they converge. For example, the exponential function 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 numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.

By recurrence

Functions whose domain are the nonnegative integers, known as sequences, 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 numbers (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. 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.

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 numbers, 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 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 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 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 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 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 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 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 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 ''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 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, 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 numbers 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 numbers, one has a function of several real variables. If the $X_i$ are equal to the set $\C$ of complex numbers, one has a function of several complex variables.

It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair 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, 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. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

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

Real function

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 numbers 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, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, that is, if 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, and whose domain is the whole real line except for 0.

The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that 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 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, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.

Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. 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, and more specifically in functional analysis, 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 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 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 functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets 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 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, 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. This theory includes the replacement axiom, which may be stated as: If is a set and is a function, then is a set.

In computer science

In computer programming, a function is, in general, a piece of a computer program, 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 is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.

Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, 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. 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, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a ''computable function'' defines also the same functions.

General recursive functions are partial 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, 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
* List of functions
* Function fitting
* Implicit function

Generalizations

* Higher-order function
* Homomorphism
* Morphism
* Microfunction
* Distribution
* Functor

Related topics

* Associative array
* Closed-form expression
* Elementary function
* Functional
* Functional decomposition
* Functional predicate
* Functional programming
* Parametric equation
* Set function
* Simple function

Notes

References

Sources

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Further reading

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

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The Wolfram Functions Site
gives formulae and visualizations of many mathematical functions.

NIST Digital Library of Mathematical Functions

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Basic concepts in set theory
Elementary mathematics