In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by , where f is the function.
More precisely, given a function , the domain of f is X. Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it.
In the special case that X and Y are both subsets of , the function f can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis.
For a function , the set Y is called the codomain, and the set of values attained by the function (which is a subset of Y) is called its range or image.
Any function can be restricted to a subset of its domain. The restriction of to , where , is written as .
If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of f. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.
Main article: Domain (mathematical analysis)
The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space where a problem is posed (i.e., where the unknown function(s) are defined).
For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y.