The inverse function f^-1 of a function f is the function for which f(f^-1(x)) = x for any x.

Given a function f(x), its inverse f^(-1)(x) is defined by f(f^(-1)(x)) = f^(-1)(f(x)) congruent x. Therefore, f(x) and f^(-1)(x) are reflections about the line y = x. In the Wolfram Language, inverse functions are represented using InverseFunction[f]. As noted by Feynman, the notation f^(-1) x is unfortunate because it conflicts with the common interpretation of a superscripted quantity as indicating a power, i.e., f^(-1) x = (1/f) x = x/f. It is therefore important to keep in mind that the symbols sin^(-1) z, cos^(-1) z, etc., refer to the inverse sine, inverse cosine, etc., and not to 1/sin z = csc z, 1/cos z = sec z, etc.

bijective | composition | inverse | inverse function theorem | inverse hyperbolic functions | inverse trigonometric functions | series reversion

InverseFunction

high school level (California Algebra II standard)