Finding and verifying inverse functions

Finding and verifying inverse functions

Reverse a function while preserving the meaning of its input, output and allowed domain. This precalculus practice set connects inverse notation with tables, reflected graphs, algebraic formulas and a calibrated measurement model.

Questions develop inverse formulas for restricted quadratic, square-root, cubic and rational functions. Each family requires a different check: a chosen branch fixes a sign, a principal root restricts the new input, and a rational rule can exclude different values from its original and inverse domains. The set also examines a proposed inverse that passes one composition check but fails the other.

Review one-to-one functions, composition, equation solving and domain restrictions before starting. Solve for the original input, interchange the roles of the variables, and identify the new domain from the original range. Then check both directions with their appropriate input sets. Inverse notation does not mean taking the reciprocal of a function value.

Single-select and multiselect items progress from reading an inverse to branch selection, verification and reversal of successive processes. Worked explanations show the decisive algebra, preserve open or excluded endpoints and explain why plausible alternatives fail. This is practice in finding and validating inverse functions; it does not claim that a few selected responses demonstrate independent proof or graph-construction mastery.