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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. In layman's terms, the domain of a function can generally be thought of as "what x can be".^{[1]}

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 , 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 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 or the complex coordinate space

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 where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

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]}

**^**"Domain, Range, Inverse of Functions".*Easy Sevens Education*. Retrieved 2023-04-13.**^**Eccles 1997, p. 91 (quote 1, quote 2); Mac Lane 1998, p. 8; Mac Lane, in Scott & Jech 1971, p. 232; Sharma 2010, p. 91; Stewart & Tall 1977, p. 89

- Bourbaki, Nicolas (1970).
*Théorie des ensembles*. Éléments de mathématique. Springer. ISBN 9783540340348. - Eccles, Peter J. (11 December 1997).
*An Introduction to Mathematical Reasoning: Numbers, Sets and Functions*. Cambridge University Press. ISBN 978-0-521-59718-0. - Mac Lane, Saunders (25 September 1998).
*Categories for the Working Mathematician*. Springer Science & Business Media. ISBN 978-0-387-98403-2. - Scott, Dana S.; Jech, Thomas J. (31 December 1971).
*Axiomatic Set Theory, Part 1*. American Mathematical Soc. ISBN 978-0-8218-0245-8. - Sharma, A. K. (2010).
*Introduction To Set Theory*. Discovery Publishing House. ISBN 978-81-7141-877-0. - Stewart, Ian; Tall, David (1977).
*The Foundations of Mathematics*. Oxford University Press. ISBN 978-0-19-853165-4.