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 [latex]f[/latex] and its inverse [latex]f^{-1}, \, f(f^{-1}(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex] and [latex]f^{-1}(f(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].
Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
The graph of a function [latex]f[/latex] and its inverse [latex]f^{-1}[/latex] are symmetric about the line [latex]y=x[/latex].

Key Equations

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

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

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

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 [latex]f[/latex] is one-to-one if [latex]f(x_1) \ne f(x_2)[/latex] if [latex]x_1 \ne x_2[/latex]