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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a functionThe words map, mapping, transformation, correspondence, and operator are often used synonymously. . from a set to a set assigns to each element of exactly one element of . The set is called the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
of the function and the set is called the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a
planet A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

planet
is a ''function'' of time.
Historically History (from Ancient Greek, Greek , ''historia'', meaning "inquiry; knowledge acquired by investigation") is the study of the past. Events occurring before the History of writing#Inventions of writing, invention of writing systems are considered ...
, the concept was elaborated with the
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
at the end of the 17th century, and, until the 19th century, the functions that were considered were
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
(that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, 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 . A function is uniquely represented by the set of all pairs , called the ''
graph of the function
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 A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

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 Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

science
, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.


Definition

A function from a
set
set
to a set is an assignment of an element of to each element of . The set is called the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
of the function and the set is called the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 A map is a symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meani ...
'' 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 In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. Formal statement In the formal language ...
, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example,
The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set. Typically, this occurs in
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, 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 In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...
function of a real variable. However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, but only that the domain is a set of real numbers that contains a non-empty
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. Such a function is then called a
partial function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
or
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of a function is the set of the
images An image (from la, imago) is an artifact that depicts visual perception, such as a photograph or other Two-dimensional space, two-dimensional picture, that resembles a subject—usually a physical physical body, object—and thus provides ...
of all elements in the domain.


Relational approach

In the relational approach, a function is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
between and that associates to each element of exactly one element of . That is, is defined by a set of ordered pairs with , such that every element of is the first component of exactly one ordered pair in . 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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of . Some authors identify it with the function; however, in common usage, the function is generally distinguished from its graph. In this approach, a function is defined as an ordered triple . In this notation, whether a function is surjective (see below) depends on the choice of . Any subset of the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of two sets and defines a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
(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
serial Serial may refer to: Arts, entertainment, and media The presentation of works in sequential segments * Serial (literature), serialised fiction in print * Serial (publishing), periodical publications and newspapers * Serial (radio and television), ...
(also called left-total) if :\forall x\in X, \exists y\in Y, \quad(x,y)\in R. A
partial function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

partial function
is a binary relation that is functional. A function is a binary relation that is functional and serial. Various properties of functions and function composition may be reformulated in the language of relations.
Gunther Schmidt Gunther Schmidt (born 1939, Rüdersdorf) is a Germans, German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhe ...
( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60,
Cambridge University Press Cambridge University Press (CUP) is the publishing business of the University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowled ...

CUP blurb for ''Relational Mathematics''
/ref> For example, a function is
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

injective
if the
converse relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is functional, where the converse relation is defined as


As an element of a Cartesian product over the domain

The set of all functions from some given domain to a codomain can be identified with the Cartesian product of copies of the codomain,
indexed Index may refer to: Arts, entertainment, and media Fictional entities * Index (A Certain Magical Index), Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo (megastr ...
by the domain. Namely, given sets and , any function is an element of the Cartesian product of copies of s over the index set : :f\in Y^X:=\prod_Y. Viewing as
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 A choice function (selector, selection) is a mathematical function ''f'' that is defined on some collection ''X'' of nonempty sets and assigns to each set ''S'' in that collection some element ''f''(''S'') of ''S''. In other words, ''f'' is a ...
.) Infinite Cartesian products are often simply "defined" as sets of functions.


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 In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is the function. More precisely, given a function f\colon X\t ...
. A more complicated example is the function :f(x)=\sin(x^2+1). In this example, the function takes a real number as input, squares it, then adds 1 to the result, then takes the sine of the result, and returns the final result as the output. When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of . Functional notation was first used by
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

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 In Latin script Latin script, also known as Roman script, is a set of graphic signs (Writing system#General properties, script) based on the letters of the classical Latin alphabet. This is derived from a form of the Cumae alphabet, Cumaean Gre ...
is customarily used instead, such as "" for the
sine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

sine function
, in contrast to italic font for single-letter symbols. When using this notation, one often encounters the
abuse of notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
whereby the notation can refer to the value of at , or to the function itself. If the variable was previously declared, then the notation unambiguously means the value of at . Otherwise, it is useful to understand the notation as being both simultaneously; this allows one to denote composition of two functions and in a succinct manner by the notation . However, distinguishing and can become important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a ''
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
''.) Other approaches of notating functions, detailed below, avoid this problem but are less commonly used.


Arrow notation

Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, x\mapsto x+1 is the function which takes a real number as input and outputs that number plus 1. Again a domain and codomain of \R is implied. The domain and codomain can also be explicitly stated, for example: :\begin \operatorname\colon \Z &\to \Z\\ x &\mapsto x^2.\end This defines a function from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f\colon X\times X\to Y;\;(x,t) \mapsto f(x,t) is a function in two variables, and we want to refer to a partially applied function X\to Y produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted x\mapsto f(x,t_0) using the arrow notation. The expression x\mapsto f(x,t_0) (read: "the map taking to ") represents this new function with just one argument, whereas the expression refers to the value of the function at the


Index notation

Index notation is often used instead of functional notation. That is, instead of writing , one writes f_x. This is typically the case for functions whose domain is the set of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. Such a function is called a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and, in this case the element f_n is called the th element of sequence. The index notation is also often used for distinguishing some variables called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

parameter
s from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map x\mapsto f(x,t) (see above) would be denoted f_t using index notation, if we define the collection of maps f_t by the formula f_t(x)=f(x,t) for all x,t\in X.


Dot notation

In the notation x\mapsto f(x), the symbol does not represent any value, it is simply a
placeholder Placeholder may refer to: Language * Placeholder name, a term or terms referring to something or somebody whose name is not known or, in that particular context, is not significant or relevant. * Filler text, text generated to fill space or provi ...
meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an
interpunct An interpunct, , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin alphabet, Latin script. (Word-separati ...

interpunct
"". This may be useful for distinguishing the function from its value at . For example, a(\cdot)^2 may stand for the function x\mapsto ax^2, and \int_a^ f(u)\,du may stand for a function defined by an integral with variable upper bound: x\mapsto \int_a^x f(u)\,du.


Specialized notations

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
and
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
,
linear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear form
s and the
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
they act upon are denoted using a
dual pair In the field of functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction i ...
to show the underlying duality. This is similar to the use of
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
in quantum mechanics. In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...
and the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
, the function notation of
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable N ...
is used to explicitly express the basic notions of function
abstraction Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...
and
application Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a co ...
. In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
, networks of functions are described in terms of how they and their compositions commute with each other using
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram with the same start an ...

commutative diagram
s 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
or ''map from to '' instead of ''
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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 Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Serge Lang
, use "function" only to refer to maps for which the
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
is a subset of the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
numbers, and use the term ''mapping'' for more general functions. In the theory of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...
s, a map denotes an
evolution function The dynamical system concept is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in So ...
used to create . See also
Poincaré map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. Whichever definition of ''map'' is used, related terms like ''
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
'', ''
codomain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

codomain
'', ''
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

injective
'', ''
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
'' 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 An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

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 In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...
that describes a combination of
arithmetic operations Arithmetic (from the Greek ἀριθμός ''arithmos'', 'number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so for ...
and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, f can be defined by the formula f(n) = n+1, for n\in\. When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s occur in the definition of a function from \mathbb to \mathbb, the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative. For example, f(x)=\sqrt defines a function f\colon \mathbb \to \mathbb whose domain is \mathbb, because 1+x^2 is always positive if is a real number. On the other hand, f(x)=\sqrt defines a function from the reals to the reals whose domain is reduced to the interval . (In old texts, such a domain was called the ''domain of definition'' of the function.) Functions are often classified by the nature of formulas that define them: *A
quadratic function In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

quadratic function
is a function that may be written f(x) = ax^2+bx+c, where are constants. *More generally, a
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
is a function that can be defined by a formula involving only additions, subtractions, multiplications, and
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 functionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is the same, with th roots and roots of polynomials also allowed. *An
elementary function In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
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
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
s and allowed.


Inverse and implicit functions

A function f\colon X\to Y, with domain and codomain , is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, if for every in , there is one and only one element in such that . In this case, the
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
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 The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
are defined this way. For example, the
cosine function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called
arccosine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

arccosine
, maps onto . The other inverse trigonometric functions are defined similarly. More generally, given a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
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 In mathematics, an implicit equation is a relation (mathematics), relation of the form , where is a function (mathematics), function of several variables (often a polynomial). For example, the implicit equation of the unit circle is . An implici ...
, because it is implicitly defined by the relation . For example, the equation of the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

unit circle
x^2+y^2=1 defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and . In this example, the equation can be solved in , giving y=\pm \sqrt, but, in more complicated examples, this is impossible. For example, the relation y^5+y+x=0 defines as an implicit function of , called the
Bring radical In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of a polynomial, root of the polynomial : x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above pol ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
provides mild
differentiability In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...
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 The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
, 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 The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

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 The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
, the exponential function, 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 of a function, graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.


Graphs and plots

Given a function f\colon X\to Y, its ''graph'' is, formally, the set :G=\. In the frequent case where and are subsets of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (or may be identified with such subsets, e.g. interval (mathematics), 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 (graphics), 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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, or the integers. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function, , is represented by a rectangle whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -axis).


General properties

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


Standard functions

There are a number of standard functions that occur frequently: * For every set , there is a unique function, called the 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 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 (mathematics), 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 property, 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 property, 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 Range may refer to: Geography * Range (geographic)A range, in geography, is a chain of hill A hill is a landform A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...
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 set, 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 (mathematics), 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[A], f^[C] 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 function, 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 function, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
'' (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 In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
, 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 factorization, 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 Nicolas Bourbaki, 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 function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
s: 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. For example, the cosine function is injective when restricted to the interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called
arccosine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

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 set union, 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 homography, homographies of the real line. A ''homography'' is a function h(x)=\frac such that . Its domain is the set of all
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s different from -d/c, and its image is the set of all real numbers different from a/c. If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting h(\infty)=a/c and h(-d/c)=\infty.


Multivariate function

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. More formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function. The
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
X_1\times\cdots\times X_n of sets X_1, \ldots, X_n is the set of all -tuples (x_1, \ldots, x_n) such that x_i\in X_i for every with 1 \leq i \leq n. Therefore, a function of variables is a function :f\colon U\to Y, where the domain has the form :U\subseteq X_1\times\cdots\times X_n. When using function notation, one usually omits the parentheses surrounding tuples, writing f(x_1,x_2) instead of f((x_1,x_2)). In the case where all the X_i are equal to the set \R of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
(see History of the function concept). At that time, only
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...
functions of a real variable were considered, and all functions were assumed to be smooth function, smooth. But the definition was soon extended to #Multivariate function, 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 In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an in ...
function of a real variable, that is, a function whose codomain is the real number, field of real numbers and whose domain is a set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s that contains an interval. In this section, these functions are simply called ''functions''. The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous function, continuous,
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
, and even analytic function, analytic. This regularity insures that these functions can be visualized by their #Graph and plots, 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 set intersection, 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 function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
s are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and
quadratic function In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

quadratic function
s. 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 The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
. A real function is monotonic function, 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 In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
, which is a real function with domain and image . This is how
inverse trigonometric functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
. Many other real functions are defined either by the
implicit function theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(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 vector (mathematics and physics), vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function. Some vector-valued functions are defined on a subset of \mathbb^n or other spaces that share geometric or topological properties of \mathbb^n, such as manifolds. These vector-valued functions are given the name ''vector fields''.


Function space

In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, and more specifically in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, a function space is a set of scalar-valued function, 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 distribution (mathematics), distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topology, topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary differential equation, 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 (mathematics), 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 In mathematics, an implicit equation is a relation (mathematics), relation of the form , where is a function (mathematics), function of several variables (often a polynomial). For example, the implicit equation of the unit circle is . An implici ...
that maps to a root of a function, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 function, 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 (set theory), class. This theory includes the Von Neumann–Bernays–Gödel set theory#NBG's axiom of replacement, 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 (programming), function is, in general, a piece of a computer program, which implementation, 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 Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable N ...
(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 computable function, 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 μ-recursive function, general recursive functions,
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable N ...
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 function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

partial function
s from integers to integers that can be defined from * constant functions, * successor function, successor, and * projection function, projection functions via the operators * #Function composition, composition, * primitive recursion, and * μ operator, minimization. Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties: * a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...), * every sequence of symbols may be coded as a sequence of bits, * a bit sequence can be interpreted as the binary representation of an integer. Lambda calculus is a theory that defines computable functions without using
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, 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
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s 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 (mathematics), Distribution * Functor


Related topics

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


Notes


References


Sources

* * * * * * *


Further reading

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


External links

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