structural:reformatting, information:extraction, structural:expansion, linguistic:translation
Definition

A function f from a set X to a set Y is an assignment of one element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function.

If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written y = f (x). {\displaystyle y=f(x).} In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the image of x under the function.

A function f, its domain X, and its codomain Y are often specified by the notation f: X → Y. {\displaystyle f:X\to Y.} One may write x ↦ y {\displaystyle x\mapsto y} instead of y = f (x) {\displaystyle y=f(x)}, where the symbol ↦ {\displaystyle \mapsto } (read ' maps to ') is used to specify where a particular element x in the domain is mapped to by f. This allows the definition of a function without naming. For example, the square function is the function x ↦ x 2. {\displaystyle x\mapsto x^{2}.}

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: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a real function, the determination of the domain of the function x ↦ 1 / f (x) {\displaystyle x\mapsto 1/f(x)} requires knowing the zeros of f. This is one of the reasons for which, in mathematical analysis, "a function from X to Y " may refer to a function having a proper subset of X as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers, typically a subset that contains a non-empty open interval. Such a function is then called a partial function.

The range or image of a function is the set of the images of all elements in the domain.

A function f on a set S means a function from the domain S, without specifying a codomain. However, some authors use it as shorthand for saying that the function is f : S → S.