With inverse function
This property is the definition of the inverse function and can hold without additional restrictions on (like , where is not ) for many named functions. In these situations, is in most cases free of branch cuts. For example, ; here means with , that is, the inverse sine function (do not confuse this with the reciprocal function ).
Some of the functions are invertible: their inversions can coincide with the original , but for other values of the parameters. For example, the inverse function for the power function is also the power function , and the relation takes place only under the restriction . In general cases the following relation takes place: .
The last property for the inverse function of the direct function can be valid under special restrictions for (where typically is not ). For example,