Range of a function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the range of a function may refer either to the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>, or the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of the function. 
In some cases the codomain and the image of a function are the same set; such a function is called <a href="/facts/Surjective_function/KezhLpCb">surjective</a> or onto. For any non-surjective function 
 
 
 
 f
 :
 X
 →
 Y
 ,
 
 
 {\displaystyle f:X\to Y,}
 
 the codomain 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and the image 
 
 
 
 
 
 
 Y
 ~
 
 
 
 
 
 {\displaystyle {\tilde {Y}}}
 
 are different; however, a new function can be defined with the original function's image as its codomain, 
 
 
 
 
 
 
 f
 ~
 
 
 
 :
 X
 →
 
 
 
 Y
 ~
 
 
 
 
 
 {\displaystyle {\tilde {f}}:X\to {\tilde {Y}}}
 
 where 
 
 
 
 
 
 
 f
 ~
 
 
 
 (
 x
 )
 =
 f
 (
 x
 )
 .
 
 
 {\displaystyle {\tilde {f}}(x)=f(x).}
 
 This new function is surjective.

Range of a function open-in-new

Range of a function