The notion of an inverse is used for many types of mathematical constructions. For example, if 
is a function restricted to a domain 
and range 
in which it is bijective and
is a function satisfying 
for all 
, then 
is the unique function with this property, called the inverse
function of 
,
written 
. It also follows that 
for all 
, so 
, i.e., inversion is symmetric. However, "inverse
functions" are also commonly defined for functions that are not bijective (most
commonly for elementary functions in the complex plane, which are multivalued),
in which case, one of both of the properties 
may fail to hold.
Inverses are also defined for elements of groups, rings, and fields (the latter two of which can possess two different types of inverses known as additive and multiplicative inverses). Every definition of inverse is symmetric and returns the starting value when applied twice.
