Summary: Inverse Functions

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, except $y=c$ 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 $y=f\left(x\right)$ , switch the variables $x$ and $y$ . Then solve for $y$ as a function of $x$ .
The graph of an inverse function is the reflection of the graph of the original function across the line $y=x$ .

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 ${f}^{-1}$