
In
mathematics, the domain or set of departure of a
function is the
set into which all of the input of the function is constrained to fall.
It is the set in the notation , and is alternatively denoted as
. Since a (total) function is defined on its entire domain, its domain coincides with its
domain of definition. However, this coincidence is no longer true for a
partial function, since the domain of definition of a partial function can be a
proper subset of the domain.
A domain is part of a function if is defined as a triple , where is called the ''domain'' of , its ''
codomain'', and its ''
graph''.
A domain is not part of a function if is defined as just a graph. For example, it is sometimes convenient in
set theory to permit the domain of a function to be a
proper class , in which case there is formally no such thing as a triple . With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form .
For instance, the domain of
cosine is the set of all
real numbers, while the domain of the
square root consists only of numbers greater than or equal to 0 (ignoring
complex numbers in both cases).
If the domain of a function is a subset of the real numbers and the function is represented in a
Cartesian coordinate system, then the domain is represented on the ''x''-axis.
Examples
A well-defined function must map every element of its domain to an element of its codomain. For example, the function
defined by
:
has no value for
. Thus the set of all
real numbers,
, cannot be its domain. In cases like this, the function is either defined on
, or the "gap is plugged" by defining
explicitly.
For example. if one extends the definition of
to the
piecewise function
:
then ''
'' is defined for all real numbers, and its domain is
.
Any function can be restricted to a subset of its domain. The
restriction of
to
, where
, is written as
.
Natural domain
The natural domain of a function (sometimes shortened as domain) is the maximum set of values for which the function is defined,
typically within the reals but sometimes among the integers or complex numbers as well. For instance, the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain, the set of possible values of the function is typically called its
range.
Category theory
Category theory deals with
morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must 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. For more, see
subobject.
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).
More common examples
As a partial function from the real numbers to the real numbers, the function
has domain
. However, if one defines the square root of a negative number ''x'' as the
complex number ''z'' with positive
imaginary part such that ''z''
2 = ''x'', then the function
has the entire real line as its domain (but now with a larger codomain).
The domain of the
trigonometric function is the set of all (real or complex) numbers, that are not of the form
.
See also
*
Attribute domain
*
Bijection, injection and surjection
*
Codomain
*
Domain decomposition
*
Effective domain
*
Image (mathematics)
*
Lipschitz domain
*
Naive set theory
*
Support (mathematics)
Notes
References
*
{{Mathematical logic
Category:Functions and mappings
Category:Basic concepts in set theory