“Function (mathematics)”的意思、由来-开放百科全书

In mathematics, a function^[1] was 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 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 {{mvar|x}} of a set {{mvar|X}}, the domain of the function, to a single element {{mvar|y}} of another set {{mvar|Y}} (possibly the same set), the codomain of the function. If the function is called {{mvar|f}}, this relation is denoted {{math|1=y = f{{space|hair}}(x)}} (read {{mvar|f}} of {{mvar|x}}), the element {{mvar|x}} is the argument or input of the function, and {{mvar|y}} is the value of the function, the output, or the image of {{mvar|x}} by {{mvar|f}}.^[2] The symbol that is used for representing the input is the variable of the function (one often says that {{mvar|f}} is a function of the variable {{mvar|x}}).

A function is uniquely represented by its graph which is the set of all pairs {{math|(x, f{{space|hair}}(x))}}. When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.{{sfn |Spivak |2008 |p=39}}

Definition

Intuitively, a function is a process that associates to each element of a set {{math|X}} a single element of a set {{math|Y}}.

Formally, a function {{math|f}} from a set {{math|X}} to a set {{math|Y}} is defined by a set {{mvar|G}} of ordered pairs {{math|(x, y)}} such that {{math|x ∈ X}}, {{math|y ∈ Y}}, and every element of {{math|X}} is the first component of exactly one ordered pair in {{mvar|G}}.^[3]^[4] In other words, for every {{math|x}} in {{math|X}}, there is exactly one element {{math|y}} such that the ordered pair {{math|(x, y)}} belongs to the set of pairs defining the function {{math|f}}. The set {{mvar|G}} is called the graph of the function. Formally speaking, 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 #Map vs function).

In the definition of function, {{math|X}} and {{math|Y}} are respectively called the domain and the codomain of the function {{mvar|f}}. If {{math|(x, y)}} belongs to the set defining {{mvar|f}}, then {{mvar|y}} is the image of {{mvar|x}} under {{mvar|f}}, or the value of {{mvar|f}} applied to the argument {{mvar|x}}. Especially in the context of numbers, one says also that {{mvar|y}} is the value of {{mvar|f}} for the value {{mvar|x}} of its variable, or, still shorter, {{mvar|y}} is the value of {{mvar|f}} of {{mvar|x}}, denoted as {{math|y {{=}} f(x)}}.

Two functions {{mvar|f}} and {{mvar|g}} are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, {{math|1=f = g}} if {{math|1=f(x) = g(x)}} for all {{math|x ∈ X}}, where {{math|f:X → Y}} and {{math|g:X → Y}}.{{sfn |Apostol |1981 |p=35}}{{sfn |Kaplan |1972 |p=25}}^[5]

The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one knows only that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function {{nowrap|from {{mvar|X}} to {{mvar|Y}} "}} often refers to a function that may have a proper subset of {{mvar|X}} as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase 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 {{mvar|f}} is a function that has the real numbers as domain and codomain, then a function mapping the value {{mvar|x}} to the value

is a function {{mvar|g}} from the reals to the reals, whose domain is the set of the reals {{mvar|x}}, such that {{math|f(x) ≠ 0}}.

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, generally in old textbooks.{{cn|reason=Give an example reference for both meanings, also to illustrate what 'old' refers to (this changes over time).|date=March 2019}}

Relational approach

Any subset of the Cartesian product of a domain

and a codomain

is said to define 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.

Univalent relations may be identified to functions whose domain is a subset of {{mvar|X}}.

Formally, functions may be identified to relations that are both univalent and left total. Violating the left-totality is similar to giving a convenient encompassing set instead of the true domain, as explained above.

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 univalent, where the converse relation is defined as

^[6]

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

, 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 x-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.)

Note: infinite Cartesian products are often simply "defined" as sets of functions.^[7]

Notation

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, often glossed over in elementary treatments of functions: functions are distinct from their values. Thus, a function {{math|f}} should be distinguished from its value {{math|f(x₀)}} at the value {{math|x₀}} in its domain. To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid the use of pedantic language. However, strictly speaking, it is an abuse of notation to write "let

be the function {{math|1=f(x) = x²}} ", since {{math|f(x)}} and {{math|x²}} should both be understood as the value of f at x, rather than the function itself. Instead, it is correct, though pedantic, to write "let

be the function defined by the equation {{math|1=f(x) = x²,}} valid for all real values of {{math|x}} ".

This distinction in language and notation becomes 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 to denoting functions, detailed below, avoid this problem but are less commonly used.

Functional notation

As first used by Leonhard Euler in 1734,^[8] functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters {{math|f, g, h}}. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). By convention, in this case, a roman type is used, such as "{{math|sin}}" for the sine function, in contrast to italic font for single-letter symbols.

means that the pair {{math|(x, y)}} belongs to the set of pairs defining the function {{mvar|f}}. If {{mvar|X}} is the domain of {{mvar|f}}, the set of pairs defining the function is thus, using set-builder notation,

Often, a definition of the function is given by what f does to the explicit argument x. For example, a function f can be defined by the equation

for all real numbers x. In this example, f 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 g and h by

and

), one of the methods below (arrow notation or dot notation) could be used.

Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise. For example,

can be written instead of

Arrow notation

For explicitly expressing domain {{math|X}} and the codomain {{math|Y}} of a function {{math|f}}, the arrow notation is often used (read: {{nowrap|"the function {{math|f}} from {{mvar|X}} to {{mvar|Y}}"}} or {{nowrap|"the function {{math|f}} mapping elements of {{mvar|X}} to elements of {{mvar|Y}}"}}):

This is often used in relation with the arrow notation for elements (read: "{{mvar|f}} maps {{mvar|x}} to {{math|f{{space|hair}}(x)}}"), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:

For example, if a multiplication is defined on a set {{mvar|X}}, then the square function

on {{mvar|X}} is unambiguously defined by (read: "the function

from {{mvar|X}} to {{mvar|X}} that maps {{mvar|x}} to {{math|x ⋅ x}}")

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 {{math|t₀}} without introducing a new function name. The map in question could be denoted

using the arrow notation for elements. Note that the expression

(read: "the map taking {{math|x}} to

") represents this new function with just one argument, whereas the expression

refers to the value of the function {{math|f}} at the {{nowrap|point

.}}

Index notation

Index notation is often used instead of functional notation. That is, instead of writing {{math|f{{space|hair}}(x)}}, 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 {{mvar|n}}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

Dot notation

the symbol {{mvar|x}} does not represent any value, it is simply a placeholder meaning that, if {{mvar|x}} is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, {{mvar|x}} may be replaced by any symbol, often an interpunct "{{math| ⋅ }}". This may be useful for distinguishing the function {{math|f{{space|hair}}(⋅)}} from its value {{math|f{{space|hair}}(x)}} at {{mvar|x}}.

For example,

may stand for the function

, and

may stand for a function defined by an integral with variable upper bound:

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.

Map vs function

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., a group homomorphism between groups can simply be called a map between those groups for the sake of succinctness. See also: maps of manifolds. Some authors^[9] reserve the word mapping to the case where the codomain Y belongs explicitly to the definition of the function. In this sense, the graph of the mapping recovers the function as the set of pairs.

Because the term "map" is synonymous with "morphism" in category theory, the term "map" can in particular emphasize the aspect that a function is a morphism in the category of sets: in the informal definition of function

, it is a subset of

consisting of all the pairs

for

. In this sense, the function doesn't capture the information of which set

is used as the co-domain. Only the range

is determined by the function.

Some authors, such as Serge Lang,^[10] use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of the fields R or C) and 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.

A partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.

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

. 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 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.

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

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 {{mvar|x}} is a real number. On the other hand,

defines a function from the reals to the reals whose domain is reduced to the interval {{math|[–1, 1]}}. (In old texts, such a domain was called the domain of definition of the function.)

Function are often classified by the nature of formulas that can that define them:

Inverse and implicit functions

A function

with domain {{mvar|X}} and codomain {{mvar|Y}}, is bijective, if for every {{mvar|y}} in {{mvar|Y}}, there is one and only one element {{mvar|x}} in {{mvar|Y}} such that {{math|1=y = f(x)}}. In this case, the inverse function of {{mvar|f}} is the function

that maps

to the element

such that {{math|1=y = f(x)}}. For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It has these 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 {{mvar|f}} to {{mvar|E}} is a bijection from {{mvar|E}} to {{mvar|F}}, 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 {{math|[0, {{pi}}]}} onto the interval {{math|[–1, 1]}}, and its inverse function, called arccosine, maps {{math|[–1, 1]}} onto {{math|[0, {{pi}}]}}. The other inverse trigonometric functions are defined similarly.

More generally, given a binary relation {{mvar|R}} between two sets {{mvar|X}} and {{mvar|Y}}, let {{mvar|E}} be a subset of {{mvar|X}} such that, for every

there is some

such that {{math|x R y}}. If one has a criterion allowing selecting such an {{mvar|y}} for every

this defines a function

called an implicit function, because it is implicitly defined by the relation {{mvar|R}}.

For example, the equation of the unit circle

defines a relation on real numbers. If {{math|–1 < x < 1}} there are two possible values of {{mvar|y}}, one positive and one negative. For {{math|1=x = ± 1}}, these two values become both equal to 0. Otherwise, there is no possible value of {{mvar|y}}. This means that the equation defines two implicit functions with domain {{math|[–1, 1]}} and respective codomains {{math|[0, +∞)}} and {{math|(–∞, 0]}}.

In this example, the equation can be solved in {{mvar|y}}, giving

but, in more complicated examples, this is impossible. For example, the relation

defines {{mvar|y}} as an implicit function of {{mvar|x}}, called the Bring radical, which has

as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and {{mvar|n}}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 {{math|1/x}} that is 0 for {{math|1=x = 1}}. An other 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 {{math|1=x = 0}}.

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.

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 (

) is a basic example, as it can be defined by the recurrence relation

Representing a function

A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps 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

In the frequent case where {{mvar|X}} and {{mvar|Y}} 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 {{math|x, y}} 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.

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

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 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 {{mvar|x}} of the domain is represented by an interval of the {{mvar|x}}-axis, and the corresponding value of the function, {{math|f(x)}}, is represented by a rectangle whose base is the interval corresponding to {{mvar|x}} and whose height is {{math|f(x)}} (possibly negative, in which case the bar extends below the {{mvar|x}}-axis).

General properties

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

Standard functions

There are a number of standard functions that occur frequently:{{anchor|Empty function}}

Function composition

Given two functions

and

such that the domain of {{mvar|g}} is the codomain of {{mvar|f}}, their composition is the function

defined by

That is, the value of

is obtained by first applying {{math|f}} to {{math|x}} to obtain {{math|1=y =f(x)}} and then applying {{math|g}} to the result {{mvar|y}} to obtain {{math|1=g(y) = g(f(x))}}. In the notation the function that is applied first is always written on the right.

The composition

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 {{math|1=f(x) = x²}} and {{math|1=g(x) = x + 1}}, 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 {{mvar|X}} to {{mvar|Y}}. That is, if {{mvar|f}} is a function with domain {{mvar|X}}, and codomain {{mvar|Y}}, one has

Image and preimage

Let

The image by {{mvar|f}} of an element {{mvar|x}} of the domain {{mvar|X}} is {{math|f(x)}}. If {{math|A}} is any subset of {{math|X}}, then the image of {{mvar|A}} by {{mvar|f}}, denoted {{math|f(A)}} is the subset of the codomain {{math|Y}} consisting of all images of elements of {{mvar|A}}, that is,

The image of {{math|f}} is the image of the whole domain, that is {{math|f(X)}}. It is also called the range of {{mvar|f}}, although the term may also refer to the codomain.^[12]

On the other hand, the inverse image, or preimage by {{mvar|f}} of a subset {{math|B}} of the codomain {{math|Y}} is the subset of the domain {{math|X}} consisting of all elements of {{math|X}} whose images belong to {{math|B}}. It is denoted by

That is

For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.

By definition of a function, the image of an element {{math|x}} of the domain is always a single element of the codomain. However, the preimage of a single element {{mvar|y}}, denoted

may be empty or contain any number of elements. For example, if {{mvar|f}} is the function from the integers to themselves that map every integer to 0, then {{math|1=f⁻¹(0) = Z}}.

is a function, {{math|A}} and {{math|B}} are subsets of {{math|X}}, and {{math|C}} and {{math|D}} are subsets of {{math|Y}}, then one has the following properties:

The preimage by {{mvar|f}} of an element {{mvar|y}} of the codomain is sometimes called, in some contexts, the fiber of {{math|y}} under {{mvar|f}}.

If a function {{mvar|f}} has an inverse (see below), this inverse is denoted

In this case

may denote either the image by

or the preimage by {{mvar|f}} of {{mvar|C}}. 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

for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

Injective, surjective and bijective functions

The function {{mvar|f}} is injective (or one-to-one, or is an injection) if {{math|f(a) ≠ f(b)}} for any two different elements {{math|a}} and {{mvar|b}} of {{mvar|X}}. Equivalently, {{mvar|f}} is injective if, for any

the preimage

contains at most one element. An empty function is always injective. If {{mvar|X}} is not the empty set, and if, as usual, the axiom of choice is assumed, then {{mvar|f}} is injective if and only if there exists a function

such that

that is, if {{mvar|f}} has a left inverse. The axiom of choice is needed, because, if {{mvar|f}} is injective, one defines {{mvar|g}} by

and by

, if

where

is an arbitrarily chosen element of {{mvar|X}}.

The function {{mvar|f}} is surjective (or onto, or is a surjection) if the range equals the codomain, that is, if {{math|1=f(X) = Y}}. In other words, the preimage

of every

is nonempty. If, as usual, the axiom of choice is assumed, then {{mvar|f}} is surjective if and only if there exists a function

such that

that is, if {{mvar|f}} has a right inverse. The axiom of choice is needed, because, if {{mvar|f}} is injective, one defines {{mvar|g}} by

where

is an arbitrarily chosen element of

The function {{mvar|f}} is bijective (or is bijection or a one-to-one correspondence) if it is both injective and surjective. That is {{mvar|f}} is bijective if, for any

the preimage

contains exactly one element. The function {{mvar|f}} is bijective if and only if it admits an inverse function, that is a function

such that

and

(Contrarily to the case of injections and surjections, this does not require the axiom of choice.)

Every function

may be factorized as the composition {{math|i ∘ s}} of a surjection followed by an injection, where {{mvar|s}} is the canonical surjection of {{mvar|X}} onto {{math|f(X)}}, and {{mvar|i}} is the canonical injection of {{math|f(X)}} into {{mvar|Y}}. This is the canonical factorization of {{mvar|f}}.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "{{math|f}} maps {{math|X}} onto {{math|Y}}" differs from "{{math|f}} maps {{math|X}} into {{math|B}}" in that the former implies that {{math|f}} is surjective, while the latter makes no assertion about the nature of {{math|f}} the mapping. 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 to be more symmetrical.

is a function, and {{mvar|S}} is a subset of {{mvar|X}}, then the restriction of {{mvar|f}} to {{mvar|S}}, denoted {{math|f_{{!}}S}}, is the function from {{mvar|S}} to {{mvar|Y}} that is defined by

This often used for define partial inverse functions: if there is a subset {{mvar|S}} of a function {{mvar|f}} such that {{math|f_{{!}}S}} is injective, then the canonical surjection of {{math|f_{{!}}S}} on its image {{math|1=f_{{!}}S(S) = f(S)}} is a bijection, which has an inverse function from {{math|f(S)}} to {{mvar|S}}. This is in this way that inverse trigonometric functions are defined. The cosine function, for example, is injective, when restricted to the interval {{math|(–0, {{pi}})}}; the image of this restriction is the interval {{math|(–1, 1)}}; this defines thus an inverse function from {{math|(–1, 1)}} to {{math|(–0, {{pi}})}}, which is called arccosine and denoted {{math|arccos}}.

Function restriction may also be used for "gluing" functions together: let

be the decomposition of {{mvar|X}} as a union of subsets. Suppose that a function

is defined on each

such that, for each pair of indices, the restrictions of

and

are equal. Then, this defines a unique function

such that

for every {{mvar|i}}. This is generally in this way that functions on manifolds are defined.

An extension of a function {{mvar|f}} is a function {{mvar|g}} such that {{mvar|f}} is a restriction of {{mvar|g}}. A typical use of this concept is the process of analytic continuation, 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. An homography is a function

such that {{math|ad – bc ≠ 0}}. Its domain is the set of all real numbers different from

and its image is the set of all real numbers different from

If one extends the real line to the projectively extended real line by adding {{math|∞}} to the real numbers, one may extend {{mvar|h}} for being a bijection of the extended real line to itself, by setting

and

Multivariate function {{anchor|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 and its speed.

More formally, a function of {{mvar|n}} variables is a function whose domain is a set of {{mvar|n}}-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

of {{mvar|n}} sets

is the set of all {{mvar|n}}-tuples

such that

for every {{mvar|i}} with

. Therefore, a function of {{mvar|n}} variables is a function

When using function notation, one usually omits the parentheses surrounding tuples, writing

instead of

In the case where all the

are equal to the set

of real numbers, one has a function of several real variables. If the

are equal to the set

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 {{math|(a, b)}} of integers with {{math|b ≠ 0}} to a pair of integers called the quotient and the remainder:

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 (see History of the function concept). 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 {{mvar|f}} and {{mvar|g}} are functions, their sum, difference and product are functions defined by

The domains of the resulting functions are the intersection of the domains of {{mvar|f}} and {{mvar|g}}. The quotient of two functions is defined similarly by

but the domain of the resulting function is obtained by removing the zeros of {{mvar|g}} from the intersection of the domains of {{mvar|f}} and {{mvar|g}}.

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

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 is differentiable in any open interval in which the original function is continuous. For example, the function

is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for {{math|1=x = 1}}, is a differentiable function called the natural logarithm.

A real function {{mvar|f}} is monotonic in an interval if the sign of

does not depend of the choice of {{mvar|x}} and {{mvar|y}} in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function {{mvar|f}} is monotonic in an interval {{mvar|I}}, it has an inverse function, which is a real function with domain {{math|f(I)}} and image {{mvar|I}}. 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

Vector-valued function

When the elements of the co-domain 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. The function that associates to each point of a fluid its velocity vector is a vector-valued function.

Some vector-valued function are defined on a subset of

or other spaces that share geometric or topological properties similar to

, like 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

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

there are two choices for the value of the square root, one of which is positive and denoted

and another which is negative and denoted

These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive {{mvar|x}}, one value for 0 and no value for negative {{mvar|x}}.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps {{mvar|y}} to a root {{mvar|x}} of

(see the figure on the right). For {{math|1=y = 0}} one may choose either

for {{mvar|x.}} By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval {{math|[–2, 2]}} and the image is {{math|[–1, 1]}}; for the second one, the domain is {{math|[–2, ∞)}} and the image is {{math|[1, ∞)}}; for the last one, the domain is {{math|(–∞, 2]}} and the image is {{math|(–∞, –1]}}. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of {{mvar|y}} that has three values for {{math|–2 < y < 2}}, and only one value for {{math|y ≤ –2}} and {{math|y ≥ –2}}.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic 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 {{mvar|i}} for the square root of –1; while, when extending through complex numbers with negative imaginary parts, one gets {{math|–i}}. There are generally two ways of solving the problem. One may define a function that is not continuous 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

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.^[13]

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 interpreted as "if {{mvar|X}} is a set, and {{mvar|F}} is a function, then {{math|F[X]}} 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 functions from integers to integers, that can be defined from the constant functions, the successor function and projection functions by mean of three operators, the composition, the primitive recursion and the minimization operators. Although defined only for functions from integers to integers, they can model any computable function, since a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...), and every sequence of symbols may be coded as a sequence of bits, which may also be viewed 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 ({{lambda}}-terms), or applications of functions to terms. Terms are manipulated through some rules, (the {{math|α}}-equivalence, the {{mvar|β}}-reduction, and the {{mvar|η}}-conversion), which are the 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 less functions than untyped lambda calculus.

Definition

Relational approach

As an element of a Cartesian product over a domain

Notation

Functional notation

Arrow notation

Index notation

Dot notation

Specialized notations

Map vs function

Specifying a function

By listing function values

By a formula

Inverse and implicit functions

Using differential calculus

By recurrence

Representing a function

Graphs and plots

Tables

Bar chart

General properties

Standard functions

Function composition

Image and preimage

Injective, surjective and bijective functions

Multivariate function {{anchor|MULTIVARIATE_FUNCTION}}

In calculus

Real function

Vector-valued function

Function space

Multi-valued functions

In the foundations of mathematics and set theory

In computer science

See also

Subpages

Generalizations

Related topics

Notes

References

Further reading

External links