Function (mathematics)

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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 the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its "input") and the other as secondary (the value of the function, or "output"). A function then is a way to associate a unique output for each input of a specified type, for example, a real number or an element of a given set. This definition covers most elementary functions, maps between algebraic structures, such as groups, and between geometric objects, such as manifolds.

One idea of enormous importance in all of mathematics is composition of functions: if y is a function of x and z is a function of y, then z is a function of x. We may describe it informally by saying that the composite function is obtained by feeding the output of the first function as the input into 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.

There are many ways to represent 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 ways apply to every possible kind of function, and one has to make a firm distinction between the function itself and multiple ways of presenting or visualizing it.

[edit] Terminology, notation, and examples

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 to each nation the name of its capital, as discussed below. Functions in algebra are usually expressible in terms of algebraic operations. Functions studied in analysis, such as the exponential function, usually have extra 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. In most mathematical fields, the terms map, mapping, and transformation are often synonymous with function. However, in some contexts they may have a more specialized meaning. In particular, 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.

Mathematical functions are frequently denoted by letters, and the standard notation for the output of a function f with the input x is f(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.

For example, the expression f(x) = x² describes a function f 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 f(3) = 9, f(1)=1, f(−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.

It is a usual practice in mathematics to introduce functions with temporary names like f; in the next paragraph we might define f(x) = 2x+1, and then f(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 one 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.

A function need not involve numbers. An example of a function that does not use numbers is the function Capital_2007 that assigns to each nation its capital in 2007 (for instance at 00:01 on 1st January). In this case Capital_2007(France) = Paris. The domain of this function is the set of world nations as of 2007, and its range is the set of world capitals. As another example, consider the function Area_of_a_triangle which is defined for all triangles in the plane and whose value on a given triangle is its area. The domain of this function is the set of all plane triangles and its range consists of positive real numbers.

[edit] Abstract definition of a function

A fairly precise, but somewhat informal definition of a function follows. Let A and B be sets. A function from A to B is determined by any association of a unique element of B with each element of A. The set A is called the domain of the function; the set B is called the codomain.

[edit] Functions versus maps

A map is usually a (total) function as defined here, especially for authors using function as synonym for partial functions with a domain (of definition) not necessarily containing all points of the source or "departure set". (For them, f:R→R; x↦1/x is a function with domain R\{0}.)

[edit] Mathematical definition of a function

The informal idea of a function as a rule has been used since ancient times and is still used as the definition of a function in informal contexts such as introductory calculus textbooks. A typical example of this informal definition, as given by Tomas and Finney (1995), is that a function is a rule that assigns to each element of a set D a single element of set C. This informal definition is sufficient for many purposes, but it relies on the undefined concept of a "rule". In the late 1800s, the question of what constitutes a valid rule defining a function came to the forefront (as described below). The consensus of modern mathematicians is that the word "rule" should be interpreted in the most general sense possible: as an arbitrary binary relation.

Thus it is common in advanced mathematics (see Bartle (2001) for an example) to formally define a function f from a set D to a set C to be a set $G f$ of ordered pairs (x,y) in the Cartesian product $D \times C$ . It is required that for each x in D there is at most one pair (x,y) in the set $G f$ . The "rule" of the function is: given x in D, if there is a pair (x,y) in $G f$ then f(x) = y, and otherwise f(x) is not defined. The set of x for which f is defined is called the domain of f; if the domain of f is all of D then f is called total and the notation $f\colon D \rightarrow C$ is used. The set C is called the codomain of the function; this must be specified because it is not determined by $G f$ . The set $G f$ is called the graph of the function.

In most areas of mathematics, the word function is used to mean total function, although non-total functions (called partial functions) are important in functional analysis, mathematical logic, and category theory.

Variations of this formal definition are sometimes more convenient for specific disciplines. In some contexts of category theory, even if a function f from a set D is not defined for every element of D, the set D is still called the domain of f. In set theory, it is common to identify the function f with its graph $G f$ ; this identification removes the need to specify either D or C in the formal definition.

[edit] History of the concept

The history of the function concept in mathematics is described by da Ponte (1992). As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope at a specific point. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis of calculus.

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. f(x) = sin(x) + x³.

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).

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.

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 independently and almost simultaneously gave the modern "formal" definition of function. In this definition, a function is a special case of a relation, in particular a function is a relation in which every first element has a unique second element.

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.

[edit] Functions in other fields

Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one quantity is completely determined by another quantity. Thus, for example, one may use a function to describe how the temperature of water affects its density.

Functions are also used in computer science to model data structures and the effects of algorithms. However, the word is also used in computing in the very different sense of procedure or sub-routine; see function (computer science).

[edit] The vocabulary of functions

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 f. The image of x may be written as f(x) or as y. (See the section on notation.)

The graph of a function f is the set of all ordered pairs (x, f(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.

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 f(A) is the subset of the range consisting of all images of elements of A. We say the f(A) is the image of A under f.

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

The preimage (or inverse image) of a subset B of the codomain Y under a function f is the subset of the domain X defined by

f⁻¹(B) = {x in X | f(x) is 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 f(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 f⁻¹(b) means f⁻¹({b}), i. e.

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 f(a) = f(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 f(x) = y; and the bijections, which are both one-to-one and onto. This nomenclature was introduced by the Bourbaki group.

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 f maps onto the set of all real numbers.

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

$g\circ f\colon X \to Z \,\!$

$x \mapsto g(f(x)) \,\!$

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 f".

Informally, the inverse of a function f is one that "undoes" the effect of f, by taking each function value f(x) to its argument x. The squaring function is the inverse of the non-negative square root function. Formally, since every function f is a relation, its inverse f⁻¹ is just the inverse relation. That is, if f has domain X, codomain Y, and graph G, the inverse has domain Y, codomain X, and graph

G⁻¹ = { (y, x) : (x, y) ∈ G }

For example, if the graph of f is G = {(1,5), (2,4), (3,5)}, then the graph of f⁻¹ is G⁻¹ = {(5,1), (4,2), (5,3)}.

The relation f⁻¹ is a function if and only if for each y in the codomain there is exactly one argument x such that f(x) = y; in other words, the inverse of a function f is a function if and only if f is a bijection. In that case, f⁻¹(f(x)) = x for every x in X, and f(f⁻¹(y)) = y for any y in Y. Sometimes a function can be modified, often by replacing the domain with a subset of the domain, and making corresponding changes in the codomain and graph, so that the modified function has an inverse that is a function.

For example, the inverse of y = sin(x), f(x) = arcsin (x), defined by y = arcsin (x) if and only if x = sin(y), is not a function, because its graph contains both the ordered pair (0, 0) and the ordered pair (0, 2π). But if we change the domain of y = sin(x) to −π/2 ≤ x ≤ π/2 and change the codomain to −1 ≤ y ≤ 1, then the resulting function does have an inverse, denoted with a capital letter A, f(x) = Arcsin (x).

This does not work with every function, however, and inverses are sometimes difficult or impossible to find.

[edit] Specifying a function

If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x).

More commonly, a function is defined by a formula, or more generally an algorithm — that is, a recipe that tells how to compute the value of f(x) given any x in the domain. More generally, a function can be defined by any mathematical condition relating the argument to the corresponding value. 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.

There is a technical sense in which most mathematical functions cannot be defined at all, in any effective way, explicit or implicit. A fundamental result of computability theory says that there are functions that can be precisely defined which cannot be computed.

[edit] 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 f, 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).

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

$f\colon \mathbb{N} \to \mathbb{R} \,\!$

$n \mapsto \frac{n}{\pi} \,\!$

Here the function named "f" 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

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

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 f is followed by g, where g produces the complex number e^ix, we may write

$\mathbb{N} \xrightarrow{f} \mathbb{R} \xrightarrow{g} \mathbb{C} . \,\!$

A more elaborate form of this is the commutative diagram.

Use of f(A) to denote the image of a subset A⊆X is consistent as 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 f[A] for the set { f(x): x ∈ A }; some authors write f`x instead of f(x), and f``A instead of f[A].

[edit] Functions with multiple inputs and outputs

[edit] Functions of two (or more) variables

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.

For example, consider the multiplication function that associates two integers to their product: f(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.

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

[edit] Functions with output in a product set

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 set.

[edit] Binary operations

The familiar binary operations of arithmetic, addition and multiplication, can be viewed as functions from R×R to R. This view is generalized in abstract algebra, where n-ary functions are used to model the operations of arbitrary algebraic structures. For example, an abstract group is defined as a set X and a function f from X×X to X that satisfies certain properties.

Traditionally, addition and multiplication are written in the infix notation: x+y and x×y instead of +(x, y) and ×(x, y).

[edit] Set of all functions

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by Y^X. The latter notation is justified by the fact that |Y^X| = |Y|^|X|. See the article on cardinal numbers for more details.

It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f is a Y-valued function of an X-valued variable". One often writes informally "Let f: X → Y" to mean "Let f be a function from X to Y".

[edit] Is a function more than its graph?

Some mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, other mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain.

There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and codomain are important mostly in formal contexts, such as category theory.

[edit] Partial functions and multi-functions

The condition for a binary relation f from X to Y to be a function can be split into two conditions:

f is total, or entire: for each x in X, there exists some y in Y such that x is related to y.
f is single-valued: for each x in X, there is at most one y in Y such that x is related to y.

Total but not single-valued

Single-valued but not total

Total and single-valued (a function)

In some contexts, a relation that satisfies condition (1), but not necessarily (2), may be called a multivalued function; and a relation that satisfies condition (2), but not necessarily (1), may be called a partial function.

[edit] 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:

[edit] Restrictions and extensions

Informally, a restriction of a function f is the result of trimming its domain.

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

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

[edit] Pointwise operations

If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows:

(f + g)(x) = f(x) + g(x)

(f × g)(x) = f(x) × g(x)

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.

[edit] Computable and non-computable functions

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. This argument shows that there are functions from integers to integers that are not computable. For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem.

[edit] Lambda calculus

The lambda calculus provides a powerful and flexible syntax for combining functions of several variables. In particular, composition becomes a special case of variable substitution; and n-ary functions can be reduced to functions with fewer arguments by a process called currying.

[edit] Functions in category theory

The notion of function can be generalized to the notion of morphism in the context of category theory. A category is a collection of objects and morphisms. The collection of objects is completely arbitrary. The morphisms are the relationships between the objects. Each morphism is an ordered triple (X, Y, f), where X is an object called the domain, Y is an object called the codomain, and f connects X to Y. There are a few restrictions on the morphisms which guarantee that they are analogous to functions (for these, see the article on categories). However, because category theory is designed to work with objects that may not be sets or may not behave like sets do, morphisms are not the same as functions.

In a concrete category, each morphism is associated with an underlying function.

[edit] See also

[edit] References

Anton, Howard (1980). Calculus with analytical geometry. John Wiley and Sons. ISBN 0-471-03248-4
Bartle, R (2001). The Elements of Real Analysis, second ed. John Wiley and Sons. ISBN 0-471-05464-X
Husch, Lawrence S. (2001). Visual Calculus. University of Tennessee.
da Ponte, João Pedro (1992). The history of the concept of function and some educational implications. The Mathematics Educator 3(2), 3-8. [1]
Thomas, G. and Finney, R (1995) Calculus and Analytic Geometry, 9th edition. Addison-Wesley. ISBN 0-201-53174-7
Godfrey Harold Hardy, (1908) A Course in Pure Mathematics, Cambridge University. ISBN 0521092272

[edit] External links

The Wolfram Functions Site, gives formulae and visualizations of many mathematical functions
Shodor: Function Flyer, interactive Java applet for graphing and exploring functions.
xFunctions, a Java applet for exploring functions graphically.
Draw Function Graphs, online drawing program for mathematical functions.
Functions from cut-the-knot
Function at ProvenMath

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Categories: Functions and mappings | Basic concepts in set theory | Elementary mathematics