Lesson 6: Inverse Relations and Functions

Property

Two functions are inverse functions if each one undoes the effect of the other.
The graphs of inverse functions are symmetric about the line $y = x$.
If we interchange the variables in the function, we get an equivalent formula for its inverse.
For example, $y = \sqrt[3]{x}$ if and only if $x = y^3$.

Examples

• The inverse of taking the fifth power of a number is taking the fifth root. If we start with $x=2$, taking the fifth power gives $2^5=32$. The fifth root of 32 is $\sqrt[5]{32}=2$, our original number.

• The functions $f(x) = x+7$ and $g(x) = x-7$ are inverses. If you take a number, say 20, then $f(20) = 27$. Applying the inverse gives $g(27) = 20$, returning to the start.

Property

When a function fails the horizontal line test, its domain can be restricted to create a one-to-one function that has an inverse function. The restricted domain should eliminate duplicate $y$-values while preserving the essential behavior of the original function.

Examples

Property

If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x, y)$, then its inverse function $f^{-1}(x)$ is the set of ordered pairs $(y, x)$.
The domain of $f$ is the range of $f^{-1}$ and the domain of $f^{-1}$ is the range of $f$.
The graphs of $f$ and $f^{-1}$ are mirror images of each other through the line $y=x$.

Examples

• The inverse of the function {(0, 1), (2, 5), (4, 9)} is the function {(1, 0), (5, 2), (9, 4)}.

• If the function $f$ has domain {0, 2, 4} and range {1, 5, 9}, its inverse $f^{-1}$ has domain {1, 5, 9} and range {0, 2, 4}.