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Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective | |||||||||||||||||||||||||||||||||

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Restriction · Composition · λ · Inverse | |||||||||||||||||||||||||||||||||

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Partial · Multivalued · Implicit | |||||||||||||||||||||||||||||||||

In mathematics, a **function**^{ [note 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.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In mathematics, a **binary relation** over two sets *A* and *B* is a set of ordered pairs consisting of elements *a* of *A* and elements *b* of *B*. That is, it is a subset of the Cartesian product *A* × *B*. It encodes the information of relation: an element *a* is related to an element *b* if and only if the pair belongs to the set.

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

- Definition
- Relational approach
- As an element of a Cartesian product over a domain
- Notation
- Functional notation
- Arrow notation
- Index notation
- Dot notation
- Specialized notations
- Other terms
- Map
- Morphism
- Specifying a function
- By listing function values
- By a formula
- Inverse and implicit functions
- Using differential calculus
- By recurrence
- Representing a function
- Graphs and plots
- Tables
- Bar chart
- General properties
- Standard functions
- Function composition
- Image and preimage
- Injective, surjective and bijective functions
- Restriction and extension
- Multivariate function
- In calculus
- Real function
- Vector-valued function
- Function space
- Multi-valued functions
- In the foundations of mathematics and set theory
- In computer science
- See also
- Subpages
- Generalizations
- Related topics
- Notes
- References
- Sources
- Further reading
- External links

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 **planet** is an astronomical body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.

The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the *x*-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.

In calculus, a **differentiable function** of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

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.^{ [1] } 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).

In mathematics, the **domain of definition** of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. Conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function.

In mathematics, an **argument** of a function is a specific input to the function; it is also called an independent variable. When it is clear from the context which argument is meant, the argument may be denoted using subscripts.

In elementary mathematics, a **variable** is a symbol, commonly a single letter, that represents a number, called the *value* of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them.

A function is uniquely represented by the set of all pairs (*x*, *f* (*x*)), called the graph of the function.^{ [note 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.

In mathematics, the **graph** of a function *f* is, formally, the set of all ordered pairs (*x*, *f* ), such that *x* is in the domain of the function *f*. In the common case where x and *f*(*x*) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the *graph of 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.^{ [2] }

**Science** is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

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.^{ [3] }^{ [note 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 mathematics, the term **mapping**, sometimes shortened to **map**, is a relationship between mathematical objects or structures.

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*).

Two functions f and g are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, *f* = *g* if *f*(*x*) = *g*(*x*) for all *x* ∈ *X*, where *f*:*X* → *Y* and *g*:*X* → *Y*.^{ [4] }^{ [5] }^{ [note 4] }

The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one knows only that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function. For example, if f is a function that has the real numbers as domain and codomain, then a function mapping the value x to the value is a function g from the reals to the reals, whose domain is the set of the reals x, such that *f*(*x*) ≠ 0.

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.^{[ citation needed ]}

Any subset of the Cartesian product of two sets and defines a binary relation between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function, given above.

A univalent or functional relation is a relation such that

for all in and in . Univalent relations may be identified with functions with codomain whose domain is a subset of X.

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.

Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is injective if the converse relation is univalent, where the converse relation is defined as ^{ [6] }

The set of all functions from some given domain to a codomain is sometimes identified with the Cartesian product of copies of the codomain, indexed by the domain. Namely, given sets and any function is an element of the Cartesian product of copies of s over the index set

Viewing as tuple with coordinates, then for each , the th coordinate of this tuple is the value This reflects the intuition that for each the function *picks* some element namely, .(This point of view is used for example in the discussion of a choice function.)

Infinite Cartesian products are often simply "defined" as sets of functions.^{ [7] }

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which defines the function using an equation that gives the names of the function and the argument explicitly. This gives rise to a subtle point, often glossed over in elementary treatments of functions: *functions* are distinct from their *values*. Thus, a function *f* should be distinguished from its value *f*(*x*_{0}) at the value *x*_{0} in its domain. To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid appearing pedantic. However, strictly speaking, it is an abuse of notation to write "let be the function *f*(*x*) = *x*^{2} ", since *f*(*x*) and *x*^{2} should both be understood as the *value* of *f* at *x*, rather than the function itself. Instead, it is correct, though long-winded, to write "let be the function defined by the equation *f*(*x*) = *x*^{2}, valid for all real values of *x* ". A compact phrasing is "let with *f*(*x*) = *x*^{2}," where the redundant "be the function" is omitted and, by convention, "for all in the domain of " is understood.

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.

As first used by Leonhard Euler in 1734,^{ [8] } 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

for all real numbers *x.* In this example, *f* can be thought of as the composite of several simpler functions: squaring, adding 1, and taking the sine. However, only the sine function has a common explicit symbol (sin), while the combination of squaring and then adding 1 is described by the polynomial expression . In order to explicitly reference functions such as squaring or adding 1 without introducing new function names (e.g., by defining function *g* and *h* by and ), one of the methods below (arrow notation or dot notation) could be used.

Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise. For example, can be written instead of

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:

For example, if a multiplication is defined on a set X, then the square function on X is unambiguously defined by (read: "the function from X to X that maps x to *x* ⋅ *x*")

the latter line being more commonly written

Often, the expression giving the function symbol, domain and codomain is omitted. Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument. As a common application of the arrow notation, suppose is a two-argument function, and we want to refer to a partially applied function produced by fixing the second argument to the value *t*_{0} without introducing a new function name. The map in question could be denoted using the arrow notation for elements. The expression (read: "the map taking *x* to ") represents this new function with just one argument, whereas the expression refers to the value of the function *f* at the point .

Index notation is often used instead of functional notation. That is, instead of writing *f* (*x*), one writes

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element is called the nth element of sequence.

The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map (see above) would be denoted using index notation, if we define the collection of maps by the formula for all .

In the notation the symbol x does not represent any value, it is simply a placeholder meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing the function *f* (⋅) from its value *f* (*x*) at x.

For example, may stand for the function , and may stand for a function defined by an integral with variable upper bound: .

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.

Term | Distinction from "function" |
---|---|

Map/Mapping | None; the terms are synonymous.^{ [9] } |

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.^{ [10] } | |

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.^{ [11] } | |

Homomorphism | A function between two structures of the same type that preserves the operations of the structure (e.g. a group homomorphism).^{ [12] }^{ [13] } |

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).^{ [14] }^{ [12] }^{ [15] } |

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^{ [16] } reserve the word *mapping* for the case where the structure of the codomain belongs explicitly to the definition of the function.

Some authors, such as Serge Lang,^{ [17] } use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term *mapping* for more general functions.

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.

Because the term "map" is synonymous with "morphism" in category theory, the term "map" can in particular emphasize the aspect that a function is a morphism in the category of sets: in the informal definition of a function , is a subset of consisting of all the pairs for . In this sense, the function doesn't capture the information that is the codomain; only the range is determined by the function.

Given a function , by definition, to each element of the domain of the function , there is a unique element associated to it, the value of at . There are several ways to specify or describe how is related to , both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function .

On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. E.g., if , then one can define a function by

Functions are often defined by a formulathat describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, can be defined by the formula , for .

When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from to the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.

For example, defines a function whose domain is because is always positive if x is a real number. On the other hand, defines a function from the reals to the reals whose domain is reduced to the interval [–1, 1]. (In old texts, such a domain was called the *domain of definition* of the function.)

Functions are often classified by the nature of formulas that 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
^{ [note 5] }is the same, with logarithms and exponential functions allowed.

A function with domain X and codomain Y, is bijective, if for every y in Y, there is one and only one element x in Y such that *y* = *f*(x). In this case, the inverse function of f is the function that maps to the element such that *y* = *f*(x). For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It has these an inverse, called the exponential function that maps the real numbers onto the positive numbers.

If a function is not bijective, it may occur that one can select subsets and such that the restriction of f to E is a bijection from E to F, and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval [0, π] onto the interval [–1, 1], and its inverse function, called arccosine, maps [–1, 1] onto [0, π]. The other inverse trigonometric functions are defined similarly.

More generally, given a binary relation R between two sets X and Y, let E be a subset of X such that, for every there is some such that *x R y*. If one has a criterion allowing selecting such an y for every this defines a function called an implicit function, because it is implicitly defined by the relation R.

For example, the equation of the unit circle defines a relation on real numbers. If –1 < *x* < 1 there are two possible values of y, one positive and one negative. For *x* = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain [–1, 1] and respective codomains [0, +∞) and (–∞, 0].

In this example, the equation can be solved in y, giving but, in more complicated examples, this is impossible. For example, the relation defines y as an implicit function of x, called the Bring radical, which has as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and nth roots.

The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.

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.

Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by . However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.

Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations.

The factorial function on the nonnegative integers () is a basic example, as it can be defined by the recurrence relation

and the initial condition

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.

Given a function its *graph* is, formally, the set

In the frequent case where X and Y are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element may be identified with a point having coordinates *x*, *y* in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the *graph of the function*. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function

consisting of all points with coordinates for yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates the plot obtained is Fermat's spiral.

A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function defined as can be represented by the familiar multiplication table

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 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).

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

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*k*^{0}= 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 id
_{X}, is the inclusion of X into itself.

Given two functions and such that the domain of g is the codomain of f, their *composition* is the function defined by

That is, the value of is obtained by first applying *f* to *x* to obtain *y* =*f*(*x*) and then applying *g* to the result y to obtain *g*(*y*) = *g*(*f*(*x*)). In the notation the function that is applied first is always written on the right.

The composition is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both and satisfy these conditions, the composition is not necessarily commutative, that is, the functions and need not be equal, but may deliver different values for the same argument. For example, let *f*(*x*) = *x*^{2} and *g*(*x*) = *x* + 1, then and agree just for

The function composition is associative in the sense that, if one of and is defined, then the other is also defined, and they are equal. Thus, one writes

The identity functions and are respectively a right identity and a left identity for functions from X to Y. That is, if f is a function with domain X, and codomain Y, one has

- A composite function
*g*(*f*(*x*)) can be visualized as the combination of two "machines". - A simple example of a function composition
- Another composition. In this example, (
*g*∘*f*)(c) = #.

Let The *image* by f of an element x of the domain X is *f*(*x*). If *A* is any subset of *X*, then the *image* of A by f, denoted *f*(*A*) is the subset of the codomain *Y* consisting of all images of elements of A, that is,

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.^{ [18] }

On the other hand, the * inverse image *, or * preimage * by f of a subset *B* of the codomain *Y* is the subset of the domain *X* consisting of all elements of *X* whose images belong to *B*. It is denoted by That is

For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.

By definition of a function, the image of an element *x* of the domain is always a single element of the codomain. However, the preimage of a single element y, denoted may be empty or contain any number of elements. For example, if f is the function from the integers to themselves that maps every integer to 0, then .

If is a function, *A* and *B* are subsets of *X*, and *C* and *D* are subsets of *Y*, then one has the following properties:

The preimage by f of an element y of the codomain is sometimes called, in some contexts, the fiber of *y* under *f*.

If a function f has an inverse (see below), this inverse is denoted In this case may denote either the image by or the preimage by f of C. This is not a problem, as these sets are equal. The notation and may be ambiguous in the case of sets that contain some subsets as elements, such as In this case, some care may be needed, for example, by using square brackets for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

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Let be a function.

The function f is * injective * (or *one-to-one*, or is an *injection*) if *f*(*a*) ≠ *f*(*b*) for any two different elements *a* and *b* of X. Equivalently, f is injective if, for any the preimage contains at most one element. An empty function is always injective. If X is not the empty set, and if, as usual, Zermelo–Fraenkel set theory is assumed, then f is injective if and only if there exists a function such that that is, if f has a left inverse. If f is injective, for defining g, one chooses an element in X (which exists as X is supposed to be nonempty),^{ [note 6] } and one defines g by if and , if

The function f is * surjective * (or *onto*, or is a *surjection*) if the range equals the codomain, that is, if *f*(*X*) = *Y*. In other words, the preimage of every is nonempty. If, as usual, the axiom of choice is assumed, then f is surjective if and only if there exists a function such that that is, if f has a right inverse. The axiom of choice is needed, because, if f is surjective, one defines g by where is an *arbitrarily chosen* element of

The function f is * bijective * (or is *bijection* or a *one-to-one correspondence*) if it is both injective and surjective. That is f is bijective if, for any the preimage contains exactly one element. The function f is bijective if and only if it admits an inverse function, that is a function such that and (Contrarily to the case of surjections, this does not require the axiom of choice.)

Every function may be factorized as the composition *i* ∘ *s* of a surjection followed by an injection, where s is the canonical surjection of X onto *f*(*X*), and i is the canonical injection of *f*(*X*) into Y. This is the *canonical factorization* of f.

"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.

If is a function and *S* is a subset of *X*, then the *restriction* of to *S*, denoted , is the function from *S* to *Y* defined by

for all *x* in *S*. Restrictions can be used to define partial inverse functions: if there is a subset *S* of the domain of a function such that is injective, then the canonical surjection of onto its image is a bijection, and thus has an inverse function from to *S*. This is in this way that inverse trigonometric functions are defined. For example, the cosine function is injective when restricted to the interval (0, π). The image of this restriction is the interval (–1, 1), and thus the restriction has an inverse function from (–1, 1) to (0, π), which is called arccosine and is denoted arccos.

Function restriction may also be used for "gluing" functions together. Let be the decomposition of X as a union of subsets, and suppose that a function is defined on each such that for each pair of indices, the restrictions of and to are equal. Then this defines a unique function such that for all i. This is the way that functions on manifolds are defined.

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.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. A *homography* is a function such that *ad* – *bc* ≠ 0. Its domain is the set of all real numbers different from and its image is the set of all real numbers different from If one extends the real line to the projectively extended real line by including ∞, one may extend h to a bijection from the extended real line to itself by setting and .

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.

The Cartesian product of n sets is the set of all n-tuples such that for every i with . Therefore, a function of n variables is a function

where the domain U has the form

When using function notation, one usually omits the parentheses surrounding tuples, writing instead of

In the case where all the are equal to the set of real numbers, one has a function of several real variables. If the are equal to the set of complex numbers, one has a function of several complex variables.

It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair (*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.

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.

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.

The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function whose graph is a hyperbola, and whose domain is the whole real line except for 0.

The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that is differentiable in any open interval in which the original function is continuous. For example, the function is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for *x* = 1, is a differentiable function called the natural logarithm.

A real function f is monotonic in an interval if the sign of does not depend of the choice of x and y in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function f is monotonic in an interval I, it has an inverse function, which is a real function with domain *f*(*I*) and image I. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.

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

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.

Some vector-valued functions are defined on a subset of or other spaces that share geometric or topological properties of , such as manifolds. These vector-valued functions are given the name *vector fields*.

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.

Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point there are several possible starting values for the function.

For example, in defining the square root as the inverse function of the square function, for any positive real number there are two choices for the value of the square root, one of which is positive and denoted and another which is negative and denoted These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive x, one value for 0 and no value for negative x.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps y to a root x of (see the figure on the right). For *y* = 0 one may choose either for x. By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [–2, 2] and the image is [–1, 1]; for the second one, the domain is [–2, ∞) and the image is [1, ∞); for the last one, the domain is (–∞, 2] and the image is (–∞, –1]. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single *multi-valued function* of y that has three values for –2 < *y* < 2, and only one value for *y* ≤ –2 and *y* ≥ –2.

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.

The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example, the singleton set may be considered as a function Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.^{ [19] }

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 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

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.

- ↑ The words
**map**,**mapping**,**transformation**,**correspondence**, and**operator**are often used synonymously. Halmos 1970 , p. 30. - ↑ This definition of "graph" refers to a
*set*of pairs of objects. Graphs, in the sense of*diagrams*, are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices). - ↑ The sets
*X*,*Y*are parts of data defining a function; i.e., a function is a set of ordered pairs with , together with the sets*X*,*Y*, such that for each , there is a unique with in the set. - ↑ This follows from the axiom of extensionality, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example, "When do two functions become equal?".
*Stack Exchange*. August 19, 2015. - ↑ Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.
- ↑ The axiom of choice is not needed here, as the choice is done in a single set.

In mathematics, a **bijection**, **bijective function**, or **one-to-one correspondence** is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function *f*: *X* → *Y* is a one-to-one (injective) and onto (surjective) mapping of a set *X* to a set *Y*. The term *one-to-one correspondence* must not be confused with *one-to-one function*.

In mathematics, a **continuous function** is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a *discontinuous* function. A continuous function with a continuous inverse function is called a homeomorphism.

In mathematics, one can often define a **direct product** of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the ancient Greek language: *ὁμός (homos)* meaning "same" and *μορφή (morphe)* meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to *ὁμός* meaning "same".

In mathematics, a **partial function** from *X* to *Y* is a function *f*: *X*′ → *Y*, for some subset *X*′ of *X*. It generalizes the concept of a function *f* : *X* → *Y* by not forcing *f* to map *every* element of *X* to an element of *Y*. If *X*′ = *X*, then *f* is called a **total function** for emphasizing that its domain is not a proper subset of *X*. Partial functions are often used when the exact domain, *X*, is not known. In real and complex analysis, a partial function is generally called simply a *function*.

In probability and statistics, a **random variable**, **random quantity**, **aleatory variable**, or **stochastic variable** is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a function *f* from a set *X* to a set *Y* is **surjective**, or a **surjection**, if for every element *y* in the codomain *Y* of *f* there is at least one element *x* in the domain *X* of *f* such that *f*(*x*) = *y*. It is not required that *x* be unique; the function *f* may map one or more elements of *X* to the same element of *Y*.

In mathematics, especially in order theory, an **upper bound** of a subset *S* of some partially ordered set is an element of *K* which is greater than or equal to every element of *S*. The term **lower bound** is defined dually as an element of *K* which is less than or equal to every element of *S*. A set with an upper bound is said to be **bounded from above** by that bound, a set with a lower bound is said to be **bounded from below** by that bound. The terms **bounded above** are also used in the mathematical literature for sets that have upper bounds.

In mathematics and in particular measure theory, a **measurable function** is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable, analogous to the definition that a function between topological spaces is continuous if it preserves the topological structure: the preimage of each open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

In mathematics, an **injective function** or **injection** or **one-to-one function** is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is the image of *at most* one element of its domain. The term *one-to-one function* must not be confused with *one-to-one correspondence*, which uniquely maps all elements in both domain and codomain to each other.

In mathematics, the **codomain** or **target set** of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the notation *f*: *X* → *Y*. The codomain is sometimes referred to as the range, but that term is ambiguous as it may also refer to the image.

In set theory and its applications to logic, mathematics, and computer science, **set-builder notation** is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.

In mathematics, and more specifically in naïve set theory, the **range** of a function refers to the *codomain* of the function, though depending upon usage it can sometimes refer to the image.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In mathematics, the **restriction of a function***f* is a new function obtained by choosing a smaller domain *A* for the original function . The notation is also used.

In complex analysis, a **complex logarithm** of the non-zero complex number z, denoted by ** w = log z**, is defined to be any complex number w for which

In mathematics, the term **fiber** can have two meanings, depending on the context:

- In naive set theory, the
**fiber**of the element*y*in the set*Y*under a map*f*:*X*→*Y*is the inverse image of the singleton under*f*. - In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.

In mathematics, a **real-valued function** is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

- ↑ MacLane, Saunders; Birkhoff, Garrett (1967).
*Algebra*(First ed.). New York: Macmillan. pp. 1–13. - ↑ Spivak 2008, p. 39.
- ↑ Hamilton, A. G. (1982).
*Numbers, sets, and axioms: the apparatus of mathematics*. Cambridge University Press. p. 83. ISBN 978-0-521-24509-8. - ↑ Apostol 1981, p. 35.
- ↑ Kaplan 1972, p. 25.
- ↑ Gunther Schmidt( 2011)
*Relational Mathematics*, Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60, Cambridge University Press ISBN 978-0-521-76268-7 CUP blurb for*Relational Mathematics* - ↑ Halmos, Naive Set Theory, 1968, sect.9 ("Families")
- ↑ Ron Larson, Bruce H. Edwards (2010),
*Calculus of a Single Variable*, Cengage Learning, p. 19, ISBN 978-0-538-73552-0 - ↑ Weisstein, Eric W. "Map".
*mathworld.wolfram.com*. Retrieved 2019-06-12. - ↑ Lang, Serge (1971),
*Linear Algebra*(2nd ed.), Addison-Wesley, p. 83 - ↑ T. M. Apostol (1981).
*Mathematical Analysis*. Addison-Wesley. p. 35. - 1 2 "function in nLab".
*ncatlab.org*. Retrieved 2019-06-12. - ↑ "homomorphism in nLab".
*ncatlab.org*. Retrieved 2019-06-12. - ↑ "morphism". nLab. Retrieved 2019-06-12.
- ↑ Weisstein, Eric W. "Morphism".
*mathworld.wolfram.com*. Retrieved 2019-06-12. - ↑ T. M. Apostol (1981).
*Mathematical Analysis*. Addison-Wesley. p. 35. - ↑ Lang, Serge (1971),
*Linear Algebra*(2nd ed.), Addison-Wesley, p. 83 - ↑
*Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology*, p. 15. ISO 80000-2 (ISO/IEC 2009-12-01) - ↑ Gödel 1940 , p. 16; Jech 2003 , p. 11; Cunningham 2016 , p. 57

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*Calculus*(4th ed.). Publish or Perish. ISBN 978-0-914098-91-1.

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*The Elements of Real Analysis*(2nd ed.). Wiley. ISBN 978-0-471-05464-1. - Dubinsky, Ed; Harel, Guershon (1992).
*The Concept of Function: Aspects of Epistemology and Pedagogy*. Mathematical Association of America. ISBN 978-0-88385-081-7. - Hammack, Richard (2009). "12. Functions" (PDF).
*Book of Proof*. Virginia Commonwealth University . Retrieved 2012-08-01. - Husch, Lawrence S. (2001).
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*The Cambridge History of Science: The modern physical and mathematical sciences*. Cambridge University Press. ISBN 978-0-521-57199-9. An approachable and diverting historical presentation. - Malik, M. A. (1980). "Historical and pedagogical aspects of the definition of function".
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Wikimedia Commons has media related to . Functions (mathematics) |

- Hazewinkel, Michiel, ed. (2001) [1994], "Function",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Function".
*MathWorld*. - The Wolfram Functions Site gives formulae and visualizations of many mathematical functions.
- NIST Digital Library of Mathematical Functions

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