- Source: 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
dom
(
f
)
{\displaystyle \operatorname {dom} (f)}
or
dom
f
{\displaystyle \operatorname {dom} f}
, 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
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
, 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
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
, 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
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
to
A
{\displaystyle A}
, where
A
⊆
X
{\displaystyle A\subseteq X}
, is written as
f
|
A
:
A
→
Y
{\displaystyle \left.f\right|_{A}\colon A\to Y}
.
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
f
{\displaystyle f}
defined by
f
(
x
)
=
1
x
{\displaystyle f(x)={\frac {1}{x}}}
cannot be evaluated at 0. Therefore, the natural domain of
f
{\displaystyle f}
is the set of real numbers excluding 0, which can be denoted by
R
∖
{
0
}
{\displaystyle \mathbb {R} \setminus \{0\}}
or
{
x
∈
R
:
x
≠
0
}
{\displaystyle \{x\in \mathbb {R} :x\neq 0\}}
.
The piecewise function
f
{\displaystyle f}
defined by
f
(
x
)
=
{
1
/
x
x
≠
0
0
x
=
0
,
{\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},}
has as its natural domain the set
R
{\displaystyle \mathbb {R} }
of real numbers.
The square root function
f
(
x
)
=
x
{\displaystyle f(x)={\sqrt {x}}}
has as its natural domain the set of non-negative real numbers, which can be denoted by
R
≥
0
{\displaystyle \mathbb {R} _{\geq 0}}
, the interval
[
0
,
∞
)
{\displaystyle [0,\infty )}
, or
{
x
∈
R
:
x
≥
0
}
{\displaystyle \{x\in \mathbb {R} :x\geq 0\}}
.
The tangent function, denoted
tan
{\displaystyle \tan }
, has as its natural domain the set of all real numbers which are not of the form
π
2
+
k
π
{\displaystyle {\tfrac {\pi }{2}}+k\pi }
for some integer
k
{\displaystyle k}
, which can be written as
R
∖
{
π
2
+
k
π
:
k
∈
Z
}
{\displaystyle \mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}}
.
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
R
n
{\displaystyle \mathbb {R} ^{n}}
or the complex coordinate space
C
n
.
{\displaystyle \mathbb {C} ^{n}.}
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
R
n
{\displaystyle \mathbb {R} ^{n}}
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.
See also
Argument of a function
Attribute domain
Bijection, injection and surjection
Codomain
Domain decomposition
Effective domain
Image (mathematics)
Lipschitz domain
Naive set theory
Range of a function
Support (mathematics)
Notes
References
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.
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