Domain of a function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the domain of a function is the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of inputs accepted by the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>. 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 <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, the function f can be graphed in the <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a>. 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 <a href="/facts/Codomain/MGc43nwQ">codomain</a>: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its <a href="/facts/Range_of_a_function/66CeTV6u">range</a> or <a href="/facts/Image_(mathematics)/KANefMAl">image</a>. 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 <a href="/facts/Restriction_(mathematics)/JV59DhKy">restriction</a> 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}
 
.

Domain of a function open-in-new

Domain of a function