Summary of Inverse Functions

For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
For a function $f$ and its inverse $f^{-1}, \, f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$ and $f^{-1}(f(x))=x$ for all $x$ in the domain of $f$ .
Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
The graph of a function $f$ and its inverse $f^{-1}$ are symmetric about the line $y=x$ .

Key Equations

Inverse functions
$f^{-1}(f(x))=x$ for all $x$ in $D$ , and $f(f^{-1}(y))=y$ for all $y$ in $R$ .

horizontal line test: a function $f$ is one-to-one if and only if every horizontal line intersects the graph of $f$ , at most, once

inverse function: for a function $f$ , the inverse function $f^{-1}$ satisfies $f^{-1}(y)=x$ if $f(x)=y$

inverse trigonometric functions: the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

one-to-one function: a function $f$ is one-to-one if $f(x_1) \ne f(x_2)$ if $x_1 \ne x_2$