The inverse image (or preimage) of a given subset 
of the codomain of 
is the set of all elements of the domain that map to the members of 
Inverse image
Let 
be a function from 
to 
The preimage or inverse image of a set 
under denoted by ![{\displaystyle f^{-1}[B],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dbf84017bfac544e2b90326d745c352343f1a1)
is the subset of defined by
![{\displaystyle f^{-1}[B]=\{x\in X\,|\,f(x)\in B\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d77c1116d2ce64b37e2423e79e466bcffb466e1)
.
Other notations include 
and 
. The inverse image of a singleton set, denoted by ![{\displaystyle f^{-1}[\{y\}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e16042ec23362592815cff817c8b2760d62671)
or by ![{\displaystyle f^{-1}[y],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc714ff09a271d825aa652c1991d34833140d64)
is also called the fiber or fiber over 
or the level set of . The set of all the fibers over the elements of 
is a family of sets indexed by .
For example, for the function 
the inverse image of 
would be 
. Again, if there is no risk of confusion, ![{\displaystyle f^{-1}[B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6089e39c5def7b9907ba317bbb0593f74e5550aa)
can be denoted by 
and 
can also be thought of as a function from the power set of to the power set of 
The notation should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of under is the image of under
