# 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].

## 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]

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**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].