Lesson 7: Inverse Functions

Property

A function is one-to-one if each value in the range corresponds to one element in the domain.
For each ordered pair in the function, each $y$-value is matched with only one $x$-value. There are no repeated $y$-values.

Horizontal Line Test
If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function.

Examples

• The set {(1, 2), (3, 4), (5, 6)} is a one-to-one function because each $y$-value (2, 4, 6) is paired with only one $x$-value.

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.