# Domain of a function

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".

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 .

## 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 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 .

## Other uses

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.

## 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*.