# Domain

In mathematics, the **domain** of a function is the set of inputs accepted by the function. It is sometimes denoted by , 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 .

## Contents

## 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

- The function defined by cannot be evaluated at 0. Therefore the natural domain of f is .
- In contrast, if is the piecewise function then
*is defined for all real numbers, and its natural domain is .* - The function has as its natural domain the non-negative real numbers, which can be denoted by , by the interval , or by .
- The tangent function has as its natural domain the set of all real numbers which are not of the form where k is any integer.

## Other uses

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*.

## Resources

- Domain Range and Toolkit Functions, Book Chapter
- Guided Notes

## Licensing

Content obtained and/or adapted from:

- Domain of a function, Wikipedia under a CC BY-SA license