Function (mathematics)
Function (mathematics)
The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve.
A function that associates any of the four colored shapes to its color.
In mathematics, a function[1] is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f (x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f.[7] The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function.[2] When the domain and the codomain are sets of real numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means to illustrate the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.[8]
Definition
This diagram, representing the set of pairs {(1,D), (2,B), (2,C)}, does not define a function. One reason is that 2 is the first element in more than one ordered pair, (2, B) and (2, C), of this set. Two other reasons, also sufficient by themselves, is that neither 3 nor 4 are first elements (input) of any ordered pair therein.
Diagram of a function, with domain X={1, 2, 3} and codomain Y={A, B, C, D}, which is defined by the set of ordered pairs {(1,D), (2,C), (3,C)}. The image/range is the set {C,D}.
Intuitively, a function is a process that associates to each element of a set X a single element of a set Y.
Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G.[9][3] In other words, for every x in X, there is exactly one element y such that the ordered pair (x, y) belongs to the set of pairs defining the function f. The set G is called the graph of the function. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions" (see section #Map).
In the definition of function, X and Y are respectively called the domain and the codomain of the function f. If (x, y) belongs to the set defining f, then y is the image of x under f, or the value of f applied to the argument x. Especially in the context of numbers, one says also that y is the value of f for the value x of its variable, or, still shorter, y is the value of f of x, denoted as y = f(x).
The range of a function is the set of the images of all elements in the domain. However, range is sometimes used as a synonym of codomain, generally in old textbooks.
Relational approach
A univalent or functional relation is a relation such that
A total relation is a relation such that
Formally, functions are identified as relations that are both univalent and total. Univalent relations that are not total are called partial functions.
As an element of a Cartesian product over a domain
Infinite Cartesian products are often simply "defined" as sets of functions.[13]
Notation
This distinction in language and notation becomes important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a functional.) Other approaches to denoting functions, detailed below, avoid this problem but are less commonly used.
Functional notation
As first used by Leonhard Euler in 1734,[14] functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters f, g, h. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). By convention, in this case, a roman type is used, such as "sin" for the sine function, in contrast to italic font for single-letter symbols.
The notation (read: "y equals f of x")
means that the pair (x, y) belongs to the set of pairs defining the function f. If X is the domain of f, the set of pairs defining the function is thus, using set-builder notation,
Often, a definition of the function is given by what f does to the explicit argument x. For example, a function f can be defined by the equation
Arrow notation
For explicitly expressing domain X and the codomain Y of a function f, the arrow notation is often used (read: "the function f from X to Y" or "the function f mapping elements of X to elements of Y"):
or
This is often used in relation with the arrow notation for elements (read: "f maps x to f (x)"), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:
the latter line being more commonly written
Index notation
Dot notation
Specialized notations
There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.
Other terms
Term | Distinction from "function" |
---|---|
Map/Mapping | None; the terms are synonymous.[15] |
A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of real or complex numbers.[16] | |
Alternatively, a map is associated with a special structure (e.g. by explicitly specifying a structured codomain in its definition). For example, a linear map.[17] | |
Homomorphism | A function between two structures of the same type that preserves the operations of the structure (e.g. a group homomorphism).[18][19] |
Morphism | A generalisation of homomorphisms to any category, even when the objects of the category are not sets (for example, a group defines a category with only one object, which has the elements of the group as morphisms; see Category (mathematics) § Examples for this example and other similar ones).[20][18][21] |
Map
A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map is often used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H). Some authors[22] reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.
Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.
Morphism
Specifying a function
By listing function values
By a formula
Functions are often classified by the nature of formulas that can that define them:
A quadratic function is a function that may be written where a, b, c are constants.
More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integers. For example, and
A rational function is the same, with divisions also allowed, such as and
An algebraic function is the same, with nth roots and roots of polynomials also allowed.
An elementary function[5] is the same, with logarithms and exponential functions allowed.
Inverse and implicit functions
The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.
Using differential calculus
Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of 1/x that is 0 for x = 1. Another common example is the error function.
More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for x = 0.
By recurrence
Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.
and the initial condition
Representing a function
A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.
Graphs and plots
The function mapping each year to its US motor vehicle death count, shown as a line chart
The same function, shown as a bar chart
Tables
y x | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
1 | 1 | 2 | 3 | 4 | 5 |
2 | 2 | 4 | 6 | 8 | 10 |
3 | 3 | 6 | 9 | 12 | 15 |
4 | 4 | 8 | 12 | 16 | 20 |
5 | 5 | 10 | 15 | 20 | 25 |
On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:
x | sin x |
---|---|
1.289 | 0.960557 |
1.290 | 0.960835 |
1.291 | 0.961112 |
1.292 | 0.961387 |
1.293 | 0.961662 |
Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.
Bar chart
Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).
General properties
This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.
Standard functions
There are a number of standard functions that occur frequently:
For every set X, there is a unique function, called the empty function from the empty set to X. The existence of the empty function from the empty set to itself is required for the category of sets to be a category – in a category, each object must have an "identity morphism", and the empty function serves as the identity for the empty set. The existence of a unique empty function from the empty set to every set A means that the empty set is an initial object in the category of sets. In terms of cardinal arithmetic, it means that k0 = 1 for every cardinal number k.
For every set X and every singleton set {s}, there is a unique function from X to {s}, which maps every element of X to s. This is a surjection (see below) unless X is the empty set.
Given a function the canonical surjection of f onto its image is the function from X to f(X) that maps x to f(x).
For every subset A of a set X, the inclusion map of A into X is the injective (see below) function that maps every element of A to itself.
The identity function on a set X, often denoted by idX, is the inclusion of X into itself.
Function composition
Image and preimage
The image of f is the image of the whole domain, that is f(X). It is also called the range of f, although the term may also refer to the codomain.[24]
- by
For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.
The preimage by f of an element y of the codomain is sometimes called, in some contexts, the fiber of y under f.
Injective, surjective and bijective functions
- (or one-to-one, or is an injection) if
- (or onto, or is a surjection) if the range equals the codomain, that is, if
- (or is bijection or a one-to-one correspondence) if it is both injective and surjective. That is
"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B" in that the former implies that f is surjective, while the latter makes no assertion about the nature of f the mapping. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.
Restriction and extension
An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.
Multivariate function
A binary operation is a typical example of a bivariate, function which assigns to each pair the result .
A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed.
More formally, a function of n variables is a function whose domain is a set of n-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function.
where the domain U has the form
It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair (a, b) of integers with b ≠ 0 to a pair of integers called the quotient and the remainder:
The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.
In calculus
The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.
Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.
Real function
Graph of a linear function
Graph of a polynomial function, here a quadratic function.
Graph of two trigonometric functions: sine and cosine.
A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.
The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.
Functions enjoy pointwise operations, that is, if f and g are functions, their sum, difference and product are functions defined by
The domains of the resulting functions are the intersection of the domains of f and g. The quotient of two functions is defined similarly by
but the domain of the resulting function is obtained by removing the zeros of g from the intersection of the domains of f and g.
Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
such that
Vector-valued function
When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function.
Function space
In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.
Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
Multi-valued functions
Together, the two square roots of all nonnegative real numbers form a single smooth curve.
Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of –1; while, when extending through complex numbers with negative imaginary parts, one gets –i. There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.
In the foundations of mathematics and set theory
The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.
These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be stated as: If X is a set and F is a function, then F[X] is a set.
In computer science
In computer programming, a function is, in general, a piece of a computer program, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every subroutine is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory.
Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else is a function that takes three functions as arguments, and, depending on the result of the first function (true or false), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below).
Except for computer-language terminology, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a computable function defines also the same functions.
General recursive functions are partial functions from integers to integers that can be defined from
constant functions,
successor, and
projection functions
via the operators
composition,
primitive recursion, and
Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:
a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...),
every sequence of symbols may be coded as a sequence of bits,
a bit sequence can be interpreted as the binary representation of an integer.
Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions (λ-terms), or applications of functions to terms. Terms are manipulated through some rules, (the α-equivalence, the β-reduction, and the η-conversion), which are the axioms of the theory and may be interpreted as rules of computation.
In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of type in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.
See also
Subpages
List of types of functions
List of functions
Function fitting
Implicit function
Generalizations
Homomorphism
Morphism
Microfunction
Distribution
Related topics
Associative array
Functional
Functional decomposition
Functional predicate
Functional programming
Parametric equation
Elementary function
Closed-form expression