№18789862[Quote]
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".[1]
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
{\displaystyle 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.[2]