Domain of a 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.
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.
- The function defined by cannot be evaluated at 0. Therefore the natural domain of is the set of real numbers excluding 0, which can be denoted by or .
- The piecewise function defined by has as its natural domain the set of real numbers.
- The square root function has as its natural domain the set of non-negative real numbers, which can be denoted by , the interval , or .
- The tangent function, denoted , has as its natural domain the set of all real numbers which are not of the form for some integer , which can be written as .
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).
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.
- Attribute domain
- Bijection, injection and surjection
- Domain decomposition
- Effective domain
- Image (mathematics)
- Lipschitz domain
- Naive set theory
- Support (mathematics)
- Weisstein, Eric W. "Domain". mathworld.wolfram.com. Retrieved 2020-08-28.
- Eccles 1997 harvnb error: no target: CITEREFEccles1997 (help), p. 91 (quote 1, quote 2); Mac Lane 1998 harvnb error: no target: CITEREFMac_Lane1998 (help), p. 8; Mac Lane, in Scott & Jech 1967 harvnb error: no target: CITEREFScottJech1967 (help), p. 232; Sharma 2004 harvnb error: no target: CITEREFSharma2004 (help), p. 91; Stewart & Tall 1977 harvnb error: no target: CITEREFStewartTall1977 (help), p. 89