In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by or, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function, the domain of f is X . 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 sets of real numbers, 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: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of f is a subset of Y , shown as the yellow oval in the accompanying diagram. Any function can be restricted to a subset of its domain. The restriction of to A, where, is written as.

Natural domain

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.

Examples

Other uses

The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty connected open subset of the real coordinate space \R^n or the complex coordinate space \C^n. Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial differential equations: in that case, a domain is the open connected subset of \R^{n} where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

Set theoretical notions

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 .