Image (mathematics)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, for a function 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
, the image of an input value 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is the single output value produced by 
 
 
 
 f
 
 
 {\displaystyle f}
 
 when passed 
 
 
 
 x
 
 
 {\displaystyle x}
 
. The preimage of an output value 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is the set of input values that produce 
 
 
 
 y
 
 
 {\displaystyle y}
 
.
More generally, evaluating 
 
 
 
 f
 
 
 {\displaystyle f}
 
 at each <a href="/facts/Element_(mathematics)/Q2mx1vHL">element</a> of a given subset 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 produces a set, called the "image of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 under (or through) 
 
 
 
 f
 
 
 {\displaystyle f}
 
". Similarly, the inverse image (or preimage) of a given subset 
 
 
 
 B
 
 
 {\displaystyle B}
 
 of the <a href="/facts/Codomain/MGc43nwQ">codomain</a> 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is the set of all elements of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that map to a member of 
 
 
 
 B
 .
 
 
 {\displaystyle B.}

The image of the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is the set of all output values it may produce, that is, the image of 
 
 
 
 X
 
 
 {\displaystyle X}
 
. The preimage of 
 
 
 
 f
 
 
 {\displaystyle f}
 
, that is, the preimage of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 under 
 
 
 
 f
 
 
 {\displaystyle f}
 
, always equals 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
); therefore, the former notion is rarely used.
Image and inverse image may also be defined for general <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a>, not just functions.

Image (mathematics) open-in-new

Image (mathematics)