Summary: Inverse Functions

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, except [latex]y=c[/latex] has an inverse. Some need a restricted domain.
  • 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 function [latex]y=f\left(x\right)[/latex], switch the variables [latex]x[/latex] and [latex]y[/latex]. Then solve for [latex]y[/latex] as a function of [latex]x[/latex].
  • The graph of an inverse function is the reflection of the graph of the original function across the line [latex]y=x[/latex].


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]