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Encyclopedia > Function (mathematics)
Graph of example function,
begin{align}&scriptstyle f colon [-1,1.5] to [-1,1.5] &textstyle x mapsto frac{(4x^3-6x^2+1)sqrt{x+1}}{3-x}end{align}

The mathematical concept of a function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output"). A function associates a single output to each input element drawn from a fixed set, such as the real numbers. In computer science, a subroutine (function, procedure, or subprogram) is a sequence of code which performs a specific task, as part of a larger program, and is grouped as one, or more, statement blocks; such code is sometimes collected into software libraries. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... In mathematics, an independent variable is any of the arguments, i. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

There are many ways to give a function: by a formula, by a plot or graph, by an algorithm that computes it, by a description of its properties. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function). In applied disciplines, functions are frequently specified by their tables of values or by a formula. Not all types of description can be given for every possible function, and one must make a firm distinction between the function itself and multiple ways of presenting or visualizing it. In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

One idea of enormous importance in all of mathematics is composition of functions: if z is a function of y and y is a function of x, then z is a function of x. We may describe it informally by saying that the composite function is obtained by using the output of the first function as the input of the second one. This feature of functions distinguishes them from other mathematical constructs, such as numbers or figures, and provides the theory of functions with its most powerful structure. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... For other uses, see Number (disambiguation). ... Look up shape in Wiktionary, the free dictionary. ...

Functions play a fundamental role in all areas of mathematics, as well as in other sciences and engineering. However, the intuition pertaining to functions, notation, and even the very meaning of the term "function" varies between the fields. More abstract areas of mathematics, such as set theory, consider very general types of functions, which may not be specified by a concrete rule and are not governed by any familiar principles. The characteristic property of a function in the most abstract sense is that it relates exactly one output to each of its admissible inputs. Such functions need not involve numbers and may, for example, associate each of a set of words with their own first letters. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

Functions in algebra are usually expressible in terms of algebraic operations. Functions studied in analysis, such as the exponential function, may have additional properties arising from continuity of space, but in the most general case cannot be defined by a single formula. Analytic functions in complex analysis may be defined fairly concretely through their series expansions. On the other hand, in lambda calculus, function is a primitive concept, instead of being defined in terms of set theory. The terms transformation and mapping are often synonymous with function. In some contexts, however, they differ slightly. In the first case, the term transformation usually applies to functions whose inputs and outputs are elements of the same set or more general structure. Thus, we speak of linear transformations from a vector space into itself and of symmetry transformations of a geometric object or a pattern. In the second case, used to describe sets whose nature is arbitrary, the term mapping is the most general concept of function. This article is about the branch of mathematics. ... In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. ... Analysis has its beginnings in the rigorous formulation of calculus. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... As the degree of the taylor series rises, it approaches the correct function. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Sphere symmetry group o. ...

Mathematical functions are denoted frequently by letters, and the standard notation for the output of a function ƒ with the input x is ƒ(x). A function may be defined only for certain inputs, and the collection of all acceptable inputs of the function is called its domain. The set of all resulting outputs is called the range of the function. However, in many fields, it is also important to specify the codomain of a function, which contains the range, but need not be equal to it. The distinction between range and codomain lets us ask whether the two happen to be equal, which in particular cases may be a question of some mathematical interest. In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all output values produced by that function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...

For example, the expression ƒ(x) = x2 describes a function ƒ of a variable x, which, depending on the context, may be an integer, a real or complex number or even an element of a group. Let us specify that x is an integer; then this function relates each input, x, with a single output, x2, obtained from x by squaring. Thus, the input of 3 is related to the output of 9, the input of 1 to the output of 1, and the input of −2 to the output of 4, and we write ƒ(3) = 9, ƒ(1)=1, ƒ(−2)=4. Since every integer can be squared, the domain of this function consists of all integers, while its range is the set of perfect squares. If we choose integers as the codomain as well, we find that many numbers, such as 2, 3, and 6, are in the codomain but not the range. The integers are commonly denoted by the above symbol. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... This picture illustrates how the hours on a clock form a group under modular addition. ... y=xÂ², for all integer values of 1â‰¤xâ‰¤25. ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ...

It is a usual practice in mathematics to introduce functions with temporary names like ƒ; in the next paragraph we might define ƒ(x) = 2x+1, and then ƒ(3) = 7. When a name for the function is not needed, often the form y = x2 is used.

If we use a function often, we may give it a more permanent name as, for example,

$operatorname{Square}(x) = x^2 . ,!$

The essential property of a function is that for each input there must be a unique output. Thus, for example, the formula

$operatorname{Root}(x) = pm sqrt x$

does not define a function of a positive real variable, because it assigns two outputs to each number: the square roots of 9 are 3 and −3. To make the square root a function, we must specify which square root to choose. The definition

$operatorname{Posroot}(x) = sqrt x ,!$

for any positive input chooses the positive square root as an output.

As mentioned above, a function need not involve numbers. By way of examples, consider the function that associates with each word its first letter or the function that associates with each triangle its area.

## Definitions

Because functions are used in so many areas of mathematics, and in so many different ways, no single definition of function has been universally adopted. Some definitions are elementary, while others use technical language that may obscure the intuitive notion. Nevertheless, the essential idea is the same in every definition.

One elementary definition, sometimes used in high school, says:

• A function pairs the elements of two sets (the domain and range) such that each member of the domain maps to exactly one element of the range

Another elementary definition may say:

• A function is given by an arithmetic expression describing how one number depends on another.

An example of such a function is y = 5x−20x3+16x5, where the value of y depends on the value of x. This is entirely satisfactory for parts of elementary mathematics, but is too clumsy and restrictive for more advanced areas. For example, the cosine function used in trigonometry cannot be written in this way; the best we can do is an infinite series, In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... In mathematics, a series is a sum of a sequence of terms. ...

$cos(x) = 1 - frac12 x^2 + frac 1{24} x^4 - frac 1{720} x^6 + dotsb$

That said, if we are willing to accept series as an extended sense of "arithmetic expression", we have a definition that served mathematics reasonably well for hundreds of years.

Eventually the gradual transformation of intuitive "calculus" into formal "analysis" brought the need for a broader definition. The emphasis shifted from how a function was presented — as a formula or rule — to a more abstract concept. Part of the new foundation was the use of sets, so that functions were no longer restricted to numbers. Thus we can say that This article is about sets in mathematics. ...

• A function ƒ from a set X to a set Y associates to each element x in X an element y = ƒ(x) in Y.

Note that X and Y need not be different sets; it is possible to have a function from a set to itself. Although it is possible to interpret the term "associates" in this definition with a concrete rule for the association, it is essential to move beyond that restriction. For example, we can sometimes prove that a function with certain properties exists, yet not be able to give any explicit rule for the association. In fact, in some cases it is impossible to give an explicit rule producing a specific y for each x, even though such a function exists. In the context of functions defined on arbitrary sets, it is not even clear how the phrase "explicit rule" should be interpreted.

As functions take on new roles and find new uses, the relationship of the function to the sets requires more precision. Perhaps every element in Y is associated with some x, perhaps not. In some parts of mathematics, including recursion theory and functional analysis, it is convenient to allow values of x with no association (in this case, the term partial function is often used). To be able to discuss such distinctions, many authors split a function into three parts, each a set: Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, a partial function is a relation that associates each element of a set (sometimes called its domain) with at most one element of another (possibly the same) set, called the codomain. ...

• A function ƒ is an ordered triple of sets (F,X,Y) with restrictions, where
F (the graph) is a set of ordered pairs (x,y),
X (the source) contains all the first elements of F and perhaps more, and
Y (the target) contains all the second elements of F and perhaps more.

The most common restrictions are that F pairs each x with just one y, and that X is just the set of first elements of F and no more.

When no restrictions are placed on F, we speak of a relation between X and Y rather than a function. The relation is "single-valued" when the first restriction holds: (x,y1)∈F and (x,y2)∈F together imply y1 = y2. Relations that are not single valued are sometimes called multivalued functions. A relation is "total" when a second restriction holds: if xX then (x,y)∈F for some y. Thus we can also say that In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

• A function from X to Y is a single-valued, total relation between X and Y.

The range of F, and of ƒ, is the set of all second elements of F; it is often denoted by rng ƒ. The domain of F is the set of all first elements of F; it is often denoted by dom ƒ. There are two common definitions for the domain of ƒ some authors define it as the domain of F, while others define it as the source of F.

The target Y of ƒ is also called the codomain of ƒ, denoted by cod ƒ; and the range of ƒ is also called the image of ƒ, denoted by im ƒ. The notation ƒ:XY indicates that ƒ is a function with domain X and codomain Y.

Some authors omit the source and target as unnecessary data. Indeed, given only the graph F, one can construct a suitable triple by taking dom F to be the source and rng F to be the target; this automatically causes F to be total. However, most authors in advanced mathematics prefer the greater power of expression afforded by the triple, especially the distinction it allows between range and codomain.

Incidentally, the ordered pairs and triples we have used are not distinct from sets; we can easily represent them within set theory. For example, we can use {{x},{x,y}} for the pair (x,y). Then for a triple (x,y,z) we can use the pair ((x,y),z). An important construction is the Cartesian product of sets X and Y, denoted by X×Y, which is the set of all possible ordered pairs (x,y) with xX and yY. We can also construct the set of all possible functions from set X to set Y, which we denote by either [XY] or YX. In mathematics, the Cartesian product is a direct product of sets. ...

We now have tremendous flexibility. By using pairs for X we can treat, say, subtraction of integers as a function, sub:Z×ZZ. By using pairs for Y we can draw a planar curve using a function, crv:RR×R. On the unit interval, I, we can have a function defined to be one at rational numbers and zero otherwise, rat:I2. By using functions for X we can consider a definite integral over the unit interval to be a function, int:[IR]→R. 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... This article deals with the concept of an integral in calculus. ...

Yet we still are not satisfied. We may want even more generality, like a function whose integral is a step function; thus we define so-called generalized functions. We may want less generality, like a function we can always actually use to get a definite answer; thus we define primitive recursive functions and then limit ourselves to those we can prove are effectively computable. Or we may want to relate not just sets, but algebraic structures, complete with operations; thus we define homomorphisms. In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. ... In mathematics, generalized functions are objects generalizing the notion of functions. ... In computability theory, primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...

## History

The word function was later used by Leonhard Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. ƒ(x) = sin(x) + x3. Euler redirects here. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... The factual accuracy of this article is disputed. ...

During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... For other uses, see Geometry (disambiguation). ... The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century. ...

At first, the idea of a function was rather limited. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. By broadening the definition of functions, mathematicians were able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion. Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Weierstrass function may also refer to the Weierstrass elliptic function () or the Weierstrass sigma, zeta, or eta functions. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

Towards the end of the 19th century, mathematicians started to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky are traditionally credited with independently giving the modern "formal" definition of a function as a relation in which every first element has a unique second element, but Dirichlet's claim to this formalization is disputed by Imre Lakatos: This article or section is in need of attention from an expert on the subject. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 â€“ May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (ÐÐ¸ÐºÐ¾Ð»Ð°ÌÐ¹ Ð˜Ð²Ð°ÌÐ½Ð¾Ð²Ð¸Ñ‡ Ð›Ð¾Ð±Ð°Ñ‡ÐµÌÐ²ÑÐºÐ¸Ð¹) (December 1, 1792â€“February 24, 1856 (N.S.); November 20, 1792â€“February 12, 1856 (O.S.)) was a Russian mathematician. ... Imre Lakatos (November 9, 1922 â€“ February 2, 1974) was a philosopher of mathematics and science. ...

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837], for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values: ...
(Proofs and Refutations, 151, Cambridge University Press 1976.)

Hardy (1908, pp. 26–28) defined a function as a relation between two variables x and y such that "to some values of x at any rate correspond values of y." He neither required the function to be defined for all values of x nor to associate each value of x to a single value of y. This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics.

The notion of a function as a rule for computing, rather than a special kind of relation, has been studied extensively in mathematical logic and theoretical computer science. Models for these computable functions include the lambda calculus, the μ-recursive functions and Turing machines. For the formal concept of computation, see computation. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are computable in some intuitive sense. ... For the test of artificial intelligence, see Turing test. ...

## Vocabulary

A specific input in a function is called an argument of the function. For each argument value x, the corresponding unique y in the codomain is called the function value at x, or the image of x under ƒ. The image of x may be written as ƒ(x) or as y. (See the section on notation.) In mathematics, the image of an element x in a set X under the function f : X â†’ Y, denoted by f(x), is the unique y in Y that is associated with x. ...

The graph of a function ƒ is the set of all ordered pairs (x, ƒ(x)), for all x in the domain X. If X and Y are subsets of R, the real numbers, then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of points. In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or left projection) and the other as the second coordinate (second entry, right projection). ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

The concept of the image can be extended from the image of a point to the image of a set. If A is any subset of the domain, then ƒ(A) is the subset of the range consisting of all images of elements of A. We say the ƒ(A) is the image of A under f. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

Notice that the range of ƒ is the image ƒ(X) of its domain, and that the range of ƒ is a subset of its codomain.

The preimage (or inverse image, or more precisely, complete inverse image) of a subset B of the codomain Y under a function ƒ is the subset of the domain X defined by In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

$f^{-1}(B) = {x in X : f(x) in B}$

So, for example, the preimage of {4, 9} under the squaring function is the set {−3,−2,+2,+3}.

In general, the preimage of a singleton set (a set with exactly one element) may contain any number of elements. For example, if ƒ(x) = 7, then the preimage of {5} is the empty set but the preimage of {7} is the entire domain. Thus the preimage of an element in the codomain is a subset of the domain. The usual convention about the preimage of an element is that ƒ−1(b) means ƒ−1({b}), i.e Generally, a singleton is something which exists alone in some way. ...

$f^{-1}(b) = {x in X : f(x) = b}$

Three important kinds of function are the injections (or one-to-one functions), which have the property that if ƒ(a) = ƒ(b) then a must equal b; the surjections (or onto functions), which have the property that for every y in the codomain there is an x in the domain such that ƒ(x) = y; and the bijections, which are both one-to-one and onto. This nomenclature was introduced by the Bourbaki group. One-to-one redirects here. ... A surjective function. ... A bijective function. ... This article is about the group of mathematicians named Nicolas Bourbaki. ...

When the first definition of function given above is used, since the codomain is not defined, the "surjection" must be accompanied with a statement about the set the function maps onto. For example, we might say ƒ maps onto the set of all real numbers.

### Restrictions and extensions

Informally, a restriction of a function ƒ is the result of trimming its domain. In mathematics, the domain of a function is the set of all input values to the function. ...

More precisely, if ƒ is a function from a X to Y, and S is any subset of X, the restriction of ƒ to S is the function ƒ|S from S to Y such that ƒ|S(s) = ƒ(s) for all s in S.

If g is any restriction of ƒ, we say that ƒ is an extension of g.

## Notation

It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus: sin x. In some formal settings, use of reverse Polish notation, x ƒ, eliminates the need for any parentheses; and, for example, the factorial function is always written n!, even though its generalization, the gamma function, is written Γ(n). Postfix notation is a mathematical notation wherein every operator follows all of its operands. ... For factorial rings in mathematics, see unique factorisation domain. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

Formal description of a function typically involves the function's name, its domain, its codomain, and a rule of correspondence. Thus we frequently see a two-part notation, an example being

begin{align} fcolon mathbb{N} &to mathbb{R} n &mapsto frac{n}{pi} end{align}

where the first part is read:

• "ƒ is a function from N to R" (one often writes informally "Let ƒ: XY" to mean "Let ƒ be a function from X to Y"), or
• "ƒ is a function on N into R", or
• "ƒ is a R-valued function of an N-valued variable",

and the second part is read:

• $n ,$ maps to $frac{n}{pi} ,!$

Here the function named "ƒ" has the natural numbers as domain, the real numbers as codomain, and maps n to itself divided by π. Less formally, this long form might be abbreviated In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

$f(n) = frac{n}{pi} , ,!$

though with some loss of information; we no longer are explicitly given the domain and codomain. Even the long form here abbreviates the fact that the n on the right-hand side is silently treated as a real number using the standard embedding.

An alternative to the colon notation, convenient when functions are being composed, writes the function name above the arrow. For example, if ƒ is followed by g, where g produces the complex number eix, we may write In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

$mathbb{N} &# 0;ghtarrow{f} mathbb{R} &# 0;ghtarrow{g} mathbb{C} . ,!$

A more elaborate form of this is the commutative diagram. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Use of ƒ(A) to denote the image of a subset AX is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g. in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is ƒ[A] for the set { ƒ(x): x ∈ A }; some authors write ƒ`x instead of ƒ(x), and ƒ``A instead of ƒ[A]. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

## Function composition

Main article: Function composition

The function composition of two or more functions uses the output of one function as the input of another. For example, ƒ(x) = sin(x2) is the composition of the sine function and the squaring function. The functions ƒ: X → Y and gY → Z can be composed by first applying ƒ to an argument x to obtain y = ƒ(x) and then applying g to y to obtain z = g(y). The composite function formed in this way from general ƒ and g may be written In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

begin{align} gcirc fcolon X &to Z x &mapsto g(f(x)). end{align}

The function on the right acts first and the function on the left acts second, reversing English reading order. We remember the order by reading the notation as "g of ƒ". The order is important, because rarely do we get the same result both ways. For example, suppose ƒ(x) = x2 and g(x) = x+1. Then g(ƒ(x)) = x2+1, while ƒ(g(x)) = (x+1)2, which is x2+2x+1, a different function.

## Identity function

Main article: Identity function

The unique function over a set X that maps each element to itself is called the identity function for X, and typically denoted by idX. Each set has its own identity function, so the subscript cannot be omitted unless the set can be inferred from context. Under composition, an identity function is "neutral": if ƒ is any function from X to Y, then An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

begin{align} f circ mathrm{id}_X &= f , mathrm{id}_Y circ f &= f . end{align}

## Inverse function

Main article: Inverse function

If ƒ is a function from X to Y then an inverse function for ƒ, denoted by ƒ−1, is a function in the opposite direction, from Y to X, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

As a simple example, if ƒ converts a temperature in degrees Celsius to degrees Fahrenheit, the function converting degrees Fahrenheit to degrees Celsius would be a suitable ƒ−1. For other uses, see Celsius (disambiguation). ... For other uses, see Fahrenheit (disambiguation). ... For other uses, see Fahrenheit (disambiguation). ... For other uses, see Celsius (disambiguation). ...

begin{align} f(C) &= tfrac95 C + 32 f^{-1}(F) &= tfrac59 (F - 32) end{align}

The notation for composition reminds us of multiplication; in fact, sometimes we denote it using juxtaposition, gƒ, without an intervening circle. Under this analogy, identity functions are like 1, and inverse functions are like reciprocals (hence the notation). Look up reciprocal in Wiktionary, the free dictionary. ...

## Specifying a function

A function can be defined by any mathematical condition relating each argument to the corresponding output value. If the domain is finite, a function ƒ may be defined by simply tabulating all the arguments x and their corresponding function values ƒ(x). More commonly, a function is defined by a formula, or (more generally) an algorithm — a recipe that tells how to compute the value of ƒ(x) given any x in the domain. In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...

There are many other ways of defining functions. Examples include recursion, algebraic or analytic closure, limits, analytic continuation, infinite series, and as solutions to integral and differential equations. The lambda calculus provides a powerful and flexible syntax for defining and combining functions of several variables. This article is about the concept of recursion. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... For other uses, see Syntax (disambiguation). ...

### Computability

Main article: computable function

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable. Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems. Computability theory is the branch of theoretical computer science that studies which problems are computationally solvable using different models of computation. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In computability theory, a busy beaver (from the colloquial expression for industrious person) is a Turing machine that, when given an initially empty (binary) tape (a string of only 0s), does a lot of work, then halts. ... In computability theory the halting problem is a decision problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... In computability theory, an undecidable problem is a problem whose language is not a recursively enumerable set. ...

### Functions with multiple inputs and outputs

The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets. In mathematics, the Cartesian product is a direct product of sets. ...

For example, consider the multiplication function that associates two integers to their product: ƒ(x, y) = x·y. This function can be defined formally as having domain Z×Z , the set of all integer pairs; codomain Z; and, for graph, the set of all pairs ((x,y), x·y). Note that the first component of any such pair is itself a pair (of integers), while the second component is a single integer. In mathematics, multiplication is an elementary arithmetic operation. ... The integers are commonly denoted by the above symbol. ...

The function value of the pair (x,y) is ƒ((x,y)). However, it is customary to drop one set of parentheses and consider ƒ(x,y) a function of two variables (or with two arguments), x and y.

The concept can still further be extended by considering a function that also produces output that is expressed as several variables. For example consider the function mirror(x, y) = (y, x) with domain R×R and codomain R×R as well. The pair (y, x) is a single value in the codomain seen as a cartesian product.

#### Binary operations

Traditionally, addition and multiplication are written in the infix notation: x+y and x×y instead of +(x, y) and ×(x, y). An infix is an affix inserted inside an existing word. ...

## Function spaces

The set of all functions from a set X to a set Y is denoted by XY, by [XY], or by YX. The latter notation is justified by the fact that |YX| = |Y||X| and is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

See the article on cardinal numbers for more details. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...

We may interpret ƒ: XY to mean ƒ ∈ [XY]; that is, "ƒ is a function from X to Y".

### Pointwise operations

If ƒ: X → R and gX → R are functions with common domain X and common codomain a ring R, then one can define the sum function ƒ + gX → R and the product function ƒ ⋅ gX → R as follows: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

begin{align} (f+g)(x) &= f(x)+g(x) , (fcdot g)(x) &= f(x) cdot g(x) , end{align}

for all x in X.

This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.

By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...

## Other properties

There are many other special classes of functions that are important to particular branches of mathematics, or particular applications. Here is a partial list:

In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... One-to-one redirects here. ... A surjective function. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... This article or section does not cite its references or sources. ... A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ... In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... A graph of the vector-valued function <2Cos(t),4Sin(t),t> A vector-valued function is a mathematical function that maps real numbers onto vectors. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ...

In mathematics, several functions or groups of functions are important enough to deserve their own names. ... A function is determined by two collections A and B and an assignment of a unique element of B to each element of A. We say this at more length, but still informally. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, the term functional is applied to certain functions. ... Functional decomposition of engineering is a method for analyzing engineered systems. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ... A plateau of a function is a part of its domain where the function has constant value. ... This article is about proportionality, the mathematical relation. ...

## References

John Wiley & Sons, Inc. ... John Wiley & Sons, Inc. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... The University of Tennessee (UT), sometimes called the University of Tennessee, Knoxville (UT Knoxville or UTK), is the flagship institution of the statewide land-grant University of Tennessee public university system in the American state of Tennessee. ... ISSN, or International Standard Serial Number, is the unique eight-digit number applied to a periodical publication including electronic serials. ... Pearson can mean Pearson PLC the media conglomerate. ...

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