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.
A function f from X to Y. The set of points in the red oval X is the domain of f.
Graph of the real-valued square root function, f(x) = √x, whose domain consists of all nonnegative real numbers
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 functionf 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 .
For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper classX, 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.
^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
Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN 9783540340348.