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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.

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*.

- 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.^{[1]} 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*.^{[2]}

**^**Weisstein, Eric W. "Domain".*mathworld.wolfram.com*. Retrieved 2020-08-28.**^**Eccles 1997 , p. 91 (quote 1, quote 2); Mac Lane 1998 , p. 8; Mac Lane, in Scott & Jech 1967 , p. 232; Sharma 2004 , p. 91; Stewart & Tall 1977 , p. 89

- Bourbaki, Nicolas (1970).
*Théorie des ensembles*. Éléments de mathématique. Springer. ISBN 9783540340348.