Inverse Function

An inverse function of a function is a function satisfying

			(1)
			(2)

for and . Such an inverse exists iff is bijective. For a real function of a real variable, the graphs of and are reflections across 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.

Examples of compositions of principal expressions are illustrated above. In complex analysis, inverse values of elementary functions are often represented by a multivalued function, even when the original is a single-valued function. A chosen branch, including the principal branch, gives an inverse on suitable restrictions of the domain and range. The two inverse identities need not hold when the corresponding principal expressions are used outside those restrictions. Determining the complete solution set of requires all appropriate inverse values, including values differing by a period for a periodic function.

In the following table, and with , and expressions are considered where defined.

An additional counterintuitive property of inverse functions is that

(3)

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