Function (mathematics)

From CryptoDox, The Online Encyclopedia on Cryptography and Information Security

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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 unique output with every input element drawn from a fixed set, such as the real numbers.

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 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.

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.

Contents 1 Introduction 2 Definition 2.1 Other definitions 2.2 Functions versus maps 3 History 4 Vocabulary 4.1 Restrictions and extensions 5 Notation 6 Function composition 7 Identity function 8 Inverse function 9 Specifying a function 9.1 Computability 9.2 Functions with multiple inputs and outputs 9.2.1 Binary operations 10 Function spaces 10.1 Pointwise operations 11 Other properties 12 See also 13 References 14 External links

Introduction

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 with each word its first letter, 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 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.

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.

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,

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

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

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.

Definition

The idea of a function as a rule has been used since ancient times and is still used as the definition of a function in contexts such as introductory calculus textbooks. A typical example of this kind of definition, as given by Template:Harvtxt, is that a function is a rule that assigns to each element of a set D one and only one element of set C. This 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.

The formal definition of function is as special kind of binary relation. The condition for a binary relation ƒ from X to Y to be a function can be split into two conditions:

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

Image:Multivalued function.svg Total but not single-valued (one input has two outputs); not a function, but sometimes called a multivalued function

Image:Partial function.svg Single-valued but not total (not defined for some elements in X); not a function, but sometimes called a partial function

Image:Total function.svg Total and single-valued relation (a function)

In some contexts, a relation that is total, but not necessarily single-valued, may be called a multivalued function; and a relation that is single-valued, but not necessarily total, may be called a partial function.

One way to define a binary relation (and hence a function) is as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of ƒ. An alternative is to define a relation as being simply the set of pairs G, without explicitly giving the domain and codomain.

There are advantages and disadvantages to each definition. The explicit domain and codomain are important in much of advanced mathematics.

Other definitions

In most areas of mathematics, the word function is used to mean total and single-valued binary relation, although partial functions are important in functional analysis, mathematical logic, and category theory, and multivalued functions are increasingly used in physics.

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 restricted domain (domain of definition) not necessarily containing all elements of the complete domain (source or "departure set"). For these authors, ƒ:R→R; x↦1/x is a function with domain R\{0}.

History

The history of the function concept in mathematics is described by Template:Harvtxt. 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. 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. ƒ(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.

Template:Harvtxt 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.

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.)

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.

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.

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) of a subset B of the codomain Y under a function ƒ is the subset of the domain X defined by

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

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.

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.

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).

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

Failed to parse (unknown error\begin): \begin{align} f\colon \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: X → Y" 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:

• maps to

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

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 e^ix, we may write

Failed to parse (unknown error\xrightarrow): \mathbb{N} \xrightarrow{f} \mathbb{R} \xrightarrow{g} \mathbb{C} . \,\!

A more elaborate form of this is the commutative diagram.

Use of ƒ(A) to denote the image of a subset A⊆X 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].

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 g: Y → 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

Failed to parse (unknown error\begin): \begin{align} g\circ f\colon 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 id_X. 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

Failed to parse (unknown error\begin): \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.

As a simple example, if ƒ converts a temperature in degrees Centigrade to degrees Fahrenheit, the function converting degrees Fahrenheit to degrees Centigrade would be a suitable ƒ⁻¹.

Failed to parse (unknown error\begin): \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).

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.

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.

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.

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.

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.

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.

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

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 ƒ 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).

Function spaces

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| and is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities.

See the article on cardinal numbers for more details.

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

Pointwise operations

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

Failed to parse (unknown error\begin): \begin{align} (f+g)(x) &= f(x)+g(x) , \\ (f\cdot 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.

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:

See also

• List of mathematical functions
• Functional
• Function composition
• Functional decomposition
• Plateau
• Proportionality

References

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External links

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• 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.io:Funciono

oc:Aplicacion (matematicas)