AP Precalculus: Inverse Functions Formulas & Concepts

1. What is an Inverse Function?

An inverse function \( f^{-1}(x) \) reverses the action of \( f(x) \):
\( f(f^{-1}(x)) = x \;\;\) and \(\;\; f^{-1}(f(x)) = x \)

- Only one-to-one functions (pass the horizontal line test) have inverses.
- The *domain* of \( f \) becomes the *range* of \( f^{-1} \) and vice versa.
- Notation: the inverse of \( f \) is written as \( f^{-1} \) (not a reciprocal!).

2. How to Find an Inverse Function Algebraically

Replace \( f(x) \) with \( y \).
Swap \( x \) and \( y \): \( x = f(y) \).
Solve the equation for \( y \) — this gives \( y = f^{-1}(x) \).
Verify by showing \( f(f^{-1}(x)) = x \).

Example:
\( f(x) = 2x + 3 \)
\( y = 2x + 3 \) → swap: \( x = 2y + 3 \) → \( y = \frac{x-3}{2} \)
So \( f^{-1}(x) = \frac{x-3}{2} \)

3. Find Values of Inverse Functions from Tables

To get a value of \( f^{-1}(x) \) from a table:

Interchange x (input) and y (output)—lookup the original output in the input column, then return the corresponding original input value.

Example Table for \( f \):

x	f(x)
1	5
2	6
4	9

\( f^{-1}(6) = 2 \) (since \( f(2) = 6 \))

4. Find Values/Graphs of Inverse Functions from Graphs

For a graph of \( f \), the graph of \( f^{-1} \) is the reflection of \( f \) over the line \( y=x \).
Every ordered pair \((a, b)\) on \( f \) becomes \((b, a)\) on \( f^{-1} \).

Example: If \( f(2) = 7 \), then \( f^{-1}(7) = 2 \).
If the graph of \( f \) passes through (2, 7), the inverse passes through (7, 2).

Key: The inverse graph always swaps all x- and y-values of the original function.

5. Find Inverse Functions & Relations

For relations that are not functions, the inverse may not be a function (it may assign the same output to multiple inputs).
Check one-to-one property: horizontal line test for functions, vertical line test for inverses.

For \( f(x) = x^2 \) (all real \( x \)), inverse is not a function. But restricting domain to \( x \geq 0 \), the inverse is \( f^{-1}(x) = \sqrt{x} \).

Inverse Function Properties & Notation

Notation: \( f^{-1} \) is inverse of \( f \). Not a reciprocal!
Composition with Original: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x))=x \) (for all \( x \) in domain/range).
Graphical Feature: The inverse graph is always a mirror over the diagonal line \( y = x \).
Domain-Range Swap: The domain of \( f \) is the range of \( f^{-1} \); the range of \( f \) is the domain of \( f^{-1} \).