# Domain

In mathematics, the **domain** of a function is the set of all input values to the function.

*X*, the set of input values, is called the domain of *f*, and *Y*, the set of **possible** output values, is called the codomain. The range of *f* is the set of all **actual** outputs {*f*(*x*) : *x* in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.

Given a function *f* : *A * → *B*, the set *A* is called the **domain**, or **domain of definition** of *f*.

The set of all values in the codomain that *f* maps to is called the range of *f*, written *f*(*A*).

A well-defined function must map every element of the domain to an element of its codomain.
For example, the function *f* defined by

*f*(*x*) = 1/*x*

has no value for *f*(0).
Thus, the set **R** of real numbers cannot be its domain.
In cases like this, the function is usually either defined on **R**\{0}, or the "gap" is plugged by specifically defining *f*(0).
If we extend the definition of *f* to

*f*(*x*) = 1/*x*, for*x*≠ 0*f*(0) = 0,

then *f* is defined for all real numbers and we can choose its domain to be **R**.

Any function can be restricted to a subset of its domain.
The restriction of *g* : *A* → *B* to *S*, where *S* ⊆ *A*, is written *g* |_{S} : *S* → *B*.

Some well-known domains are as follows (note that each successive domain includes those above it):

Natural numbers | 1,2,3,4 | |

Whole numbers | 0 | |

Integers | -1,-2,-3,-4 | |

Rational numbers | 1/3, 1/985 | |

Real numbers | ||

Complex numbers |

## Category theory

In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to be abandoned, or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

## Complex analysis

In complex analysis, a domain is an open connected subset of the complex numbers.

## See also

cs:Definiční obor da:Definitionsmængde de:Definitionsmenge et:Määramispiirkond es:Dominio de definición fr:Ensemble de définition io:Ensemblo di defino is:Skilgreiningarmengi nl:Domein (wiskunde) pl:Dziedzina sv:Definitionsmängd zh:定义域