Key Concepts

• If [latex]g\left(x\right)[/latex] is the inverse of [latex]f\left(x\right)[/latex], then
• [latex]g\left(f\left(x\right)\right)=f\left(g\left(x\right)\right)=x[/latex].
• Each of the toolkit functions has an inverse.
• For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
• A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
• For a tabular function, exchange the input and output rows to obtain the inverse.
• The inverse of a function can be determined at specific points on its graph.
• To find the inverse of a formula, solve the equation [latex]y=f\left(x\right)[/latex] for [latex]x[/latex] as a function of [latex]y[/latex]. Then exchange the labels [latex]x[/latex] and [latex]y[/latex].
• The graph of an inverse function is the reflection of the graph of the original function across the line [latex]y=x[/latex].

Glossary

inverse function

for any one-to-one function [latex]f\left(x\right)[/latex], the inverse is a function [latex]{f}^{-1}\left(x\right)[/latex] such that [latex]{f}^{-1}\left(f\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]; this also implies that [latex]f\left({f}^{-1}\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]{f}^{-1}[/latex]