Function (mathematics)

In mathematics, a function from a set X exactly one element of Y is called the codomain of the function.

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 a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept.

A function is often denoted by a letter such as h. The value of a function x) is denoted by x = 4 is denoted by f(x)=x2+1; in this case, some computation, called function evaluation, may be needed for deducing the value of the function at a particular value; for example, if (x, f (x)), called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.

Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details.

A specific element Y is the value of the function at f, its domain f:X→Y. One may write ↦ (read 'maps to') is used to specify where a particular element x↦x2.

The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if f. This is one of the reasons for which, in mathematical analysis, "a function X to S means a function from the domain X and x∈X and Y and denoted Y is a binary relation Y that satisfies the two following conditions:

• For every y in (x,y)∈R and X, the codomain R⊆{(x,y)∣x∈X,y∈Y}
• ⁠R⁠. For every ⁠x⁠ is denoted ⁠y=f(x)⁠ instead of ⁠⁠y⁠", "⁠x⁠", or "the application of ⁠y⁠", etc.

  Partial functions

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  X to X and y in x∈X, either X such that X to X, one often says that the partial function is a total function.

  In several areas of mathematics, the term "function" refers to partial functions rather than to ordinary (total) functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.

  In calculus, a real-valued function of a real variable or real function is a partial function from the set x↦1/f(x) is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse.

  Similarly, a function of a complex variable is generally a partial function whose domain of definition is a subset of the complex numbers 0 belongs to its domain of definition (see Halting problem).

  Multivariate functions

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  (x,y) the result n-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a parametric surface, as used in, e.g., bivariate interpolation.

  Commonly, an f(x1,…,xn) instead of X1,…,Xn, the set of all x1∈X1,…,xn∈Xn is called the Cartesian product of f:U→Y,

  where the domain Xi are equal to the set f. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in

  3 in the above example), or an expression that can be evaluated to an element of the domain (sin" for the sine function, in contrast to italic font for single-letter symbols.

  The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let sqr⁡:ℤ→ℤx↦x2.

  This defines a function X→Y produced by fixing the second argument to the value 0</sub>"}},"i":0}}]}'/>t0 without introducing a new function name. The map in question could be denoted x to t nought") represents this new function with just one argument, whereas the expression 0</sub>, ''t''₀)"}},"i":0}}]}'/>f(x0, t0) refers to the value of the function (x0, t0).

  Index notation

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  Index notation may be used instead of functional notation. That is, instead of writing fn is called the ft using index notation, if we define the collection of maps x,t∈X.

  Placeholder notation

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  In the notation 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, in the expression to the right of the arrow, f or a(−)2 may stand for the function ∫a(−)f(u)du may stand for a function defined by an integral with variable upper bound: (x,y) to the sum of their squares, f(w,x,y),

  Term	Distinction from "function"
  Map/Mapping	None; the terms are synonymous.
  	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.
  	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.
  Homomorphism	A function between two structures of the same type that preserves the operations of the structure (e.g. a group homomorphism).
  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).

  A function may also be 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 may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to x of the domain of the function f at f(x), 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 f:A→ℝ by f(n)=n+1, for ℝ, 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, ℝ, because f(x)=1−x2 defines a function from the reals to the reals whose domain is reduced to the interval 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 integer powers. For example, x.
• A rational function is the same, with divisions also allowed, such as nth roots and roots of polynomials also allowed.
• An elementary function is the same, with logarithms and exponential functions allowed.