, a function
[The words map, mapping, transformation, correspondence, and operator are often used synonymously. .]
is a binary relation
between two sets
that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integer
s to integers, or from the real number
s to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet
is a ''function'' of time. 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 a process or a relation that associates each element of a set
, the ''domain
'' of the function, to a single element of another set (possibly the same set), the ''codomain
'' of the function. It is customarily denoted by letters such as , and .
If the function is called , this relation is denoted by (which reads " of "), where the element is the ''argument
'' or ''input'' of the function, and is the ''value of the function'', the ''output'', or the ''image'' of by .
The symbol that is used for representing the input is the variable
of the function (e.g., is a function of the variable ).
A function is uniquely represented by the set of all pairs
, called the graph
of 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. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science
, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
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Intuitively, a function is a process that associates each element of a set , to a single element of a set .
Formally, a function from a set to a set is defined by a set of ordered pairs with , such that every element of is the first component of exactly one ordered pair in .
[The sets , are usually parts of data defining a function; i.e., a function is a set together with the sets , . For example, the same may lead to a surjective (see below) and a non-surjective function, depending on .]
In other words, for every in , there is exactly one element such that the ordered pair belongs to the set of pairs defining the function . The set is called the graph of the function
. Occasionally, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph.
Functions are also called ''maps
'' or ''mappings'', though some authors make some distinction between "maps" and "functions" (see section #Map
The fact of being a function from the set to the set is formally denoted by .
In the definition of a function, and are respectively called the ''domain'' and the ''codomain'' of the function . If belongs to the set defining , then is the ''image'' of under , or the ''value'' of applied to the ''argument'' . In the context of numbers in particular, one also says that is the value of for the ''value of its variable'', or, more concisely, that is the ''value of'' ''of'' , denoted as .
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, if for all , where and .
[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 subset
[called 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
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 of a function
is the set of the images
of all elements in the domain.
However, ''range'' is sometimes used as a synonym of codomain,
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 functional
(also called right-unique) if
A binary relation is serial
(also called left-total) if
A partial function
is a binary relation that is functional.
A function is a binary relation that is functional and serial.
The functional property is also commonly referred to as the function being well defined.
Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is injective
if the converse relation
is functional, where the converse relation is defined as .
[Gunther Schmidt( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60, Cambridge University Press ]
CUP blurb for ''Relational Mathematics''
As an element of a Cartesian product over a domain
The set of all functions from some given domain to a codomain is sometimes identified with the Cartesian product of copies of the codomain, indexed by the domain. Namely, given sets and , any function is an element of the Cartesian product of copies of s over the index set
Viewing as tuple with coordinates, then for each , the th coordinate of this tuple is the value . This reflects the intuition that for each , the function ''picks'' some element , namely, . (This point of view is used for example in the discussion of a choice function.)
Infinite Cartesian products are often simply "defined" as sets of functions.
There are various standard ways for denoting functions. The most commonly used notation is functional notation, which defines the function using an equation that gives the names of the function and the argument explicitly. This gives rise to a subtle point which is often glossed over in elementary treatments of functions: ''functions'' are distinct from their ''values''. Thus, a function should be distinguished from its value at the value in its domain. To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid appearing pedantic. However, strictly speaking, it is an abuse of notation to write "let be the function ", since and should both be understood as the ''value'' of ''f'' at ''x'', rather than the function itself. Instead, it is correct, though long-winded, to write "let be the function defined by the equation valid for all real values of ". A compact phrasing is "let with " where the redundant "be the function" is omitted and, by convention, "for all in the domain of " is understood.
This distinction in language and notation 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.
As first used by Leonhard Euler in 1734, functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters .
Some widely-used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In which case, a roman type is customarily used instead, such as "" for the sine function, in contrast to italic font for single-letter symbols.
The notation (read: " equals of ")
means that the pair belongs to the set of pairs defining the function . If is the domain of , the set of pairs defining the function is thus, using set-builder notation,
Often, a definition of the function is given by what does to the explicit argument . For example, a function can be defined by the equation
for all real numbers . In this example, can be thought of as the composite of several simpler functions: squaring, adding 1, and taking the sine. However, only the sine function has a common explicit symbol (sin), while the combination of squaring and then adding 1 is described by the polynomial expression . In order to explicitly reference functions such as squaring or adding 1 without introducing new function names (e.g., by defining function and by and ), one of the methods below (arrow notation or dot notation) could be used.
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 .
For explicitly expressing domain and the codomain of a function , the arrow notation is often used (read: or ):
This is often used in relation with the arrow notation for elements (read: " maps to "), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:
For example, if a multiplication is defined on a set , then the square function on is unambiguously defined by (read: "the function from to that maps to ")
the latter line being more commonly written
Often, the expression giving the function symbol, domain and codomain is omitted. Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument. As a common application of the arrow notation, suppose is a two-argument function, and we want to refer to a partially applied function produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted using the arrow notation for elements. The expression (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 is often used instead of functional notation. That is, instead of writing , one writes
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 is called the th element of 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 (see above) would be denoted using index notation, if we define the collection of maps by the formula for all .
In the notation
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, may stand for the function , and may stand for a function defined by an integral with variable upper bound: .
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.
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 , by definition, to each element of the domain of the function , there is a unique element associated to it, the value of at . There are several ways to specify or describe how is related to , 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 .
By listing function values
On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. E.g., if , then one can define a function by
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, can be defined by the formula , for .
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 to 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, defines a function whose domain is because is always positive if is a real number. On the other hand, 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 can that define them:
*A quadratic function is a function that may be written 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, and
*A rational function is the same, with divisions also allowed, such as and
*An algebraic function is the same, with th roots and roots of polynomials also allowed.
*An elementary function
[Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.] is the same, with logarithms and exponential functions allowed.
Inverse and implicit functions
A function 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 that maps to the element 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 is not bijective, it may occur that one can select subsets and 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 there is some such that . If one has a criterion allowing selecting such an for every this defines a function called an implicit function, because it is implicitly defined by the relation .
For example, the equation of the unit circle 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 but, in more complicated examples, this is impossible. For example, the relation defines as an implicit function of , called the Bring radical, which has 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 . 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.
Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.
The factorial function on the nonnegative integers () is a basic example, as it can be defined by the recurrence relation
and the initial condition
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 its ''graph'' is, formally, the set
In the frequent case where and are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element 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
consisting of all points with coordinates for yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates the plot obtained is Fermat's spiral.
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 defined as 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 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).
This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.
There are a number of standard functions that occur frequently:
* For every set , there is a unique function, called the 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 the empty function is a convention that is needed for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements.
* 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 the ''canonical surjection'' of onto its image 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.
Given two functions and such that the domain of is the codomain of , their ''composition'' is the function defined by
That is, the value of 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 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 and satisfy these conditions, the composition is not necessarily commutative, that is, the functions and need not be equal, but may deliver different values for the same argument. For example, let and , then and agree just for
The function composition is associative in the sense that, if one of and is defined, then the other is also defined, and they are equal. Thus, one writes
The identity functions and 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
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 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,
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 and is given by the equation
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 and is given by the equation
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 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 .
If is a function, and are subsets of , and and are subsets of , then one has the following properties:
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 In this case may denote either the image by or the preimage by of . This is not a problem, as these sets are equal. The notation and 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