Inverse Function

Given a function , its inverse is defined by

(1)

Therefore, and are reflections about the line . In the Wolfram Language, inverse functions are represented using InverseFunction[f].

As noted by Feynman (1997), the notation is unfortunate because it conflicts with the common interpretation of a superscripted quantity as indicating a power, i.e., . It is therefore important to keep in mind that the symbols , , etc., refer to the inverse sine, inverse cosine, etc., and not to , , etc.

A function admits an inverse function (i.e., " is invertible") iff it is bijective. However, inverse functions are commonly defined for elementary functions that are multivalued in the complex plane. In such cases, the inverse relation holds on some subset of the complex plane but, over the whole plane, either or both parts of the identity may fail to hold. A few examples are illustrated above and in the following table. In the table and .

An additional counterintuitive property of inverse functions is that

(2)

so the expected identity does not hold along the negative real axis.