Key Concepts

• If $g\left(x\right)$ is the inverse of $f\left(x\right)$, then
• $g\left(f\left(x\right)\right)=f\left(g\left(x\right)\right)=x$.
• 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 $y=f\left(x\right)$ for $x$ as a function of $y$. Then exchange the labels $x$ and $y$.
• The graph of an inverse function is the reflection of the graph of the original function across the line $y=x$.

Glossary

inverse function

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