Inverse Functions
By the end of this lesson, you’ll be able to:
- Understand what an inverse function represents.
- Explain how a function and its inverse “undo” each other.
- Determine when a function does or does not have an inverse.
Key Ideas
An inverse function reverses the action of the original function.
If \(f\) has an inverse, then:
\[ f\bigl(f^{-1}(x)\bigr) = x \qquad\text{and}\qquad f^{-1}(f(x)) = x. \]
In other words, a function and its inverse undo each other.
If a function maps:
\[ 3 \rightarrow 7, \]
then its inverse maps:
\[ 7 \rightarrow 3. \]
Visually, the graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line:
\[ y = x. \]
A function has an inverse only if it is one-to-one, meaning each output comes from exactly one input.
A quick way to check this is the horizontal line test.
If every horizontal line intersects the graph at most once, the function is one-to-one and has an inverse.
Common Problem Types
1. Evaluating Inverses From Given Information
Using the fact that inverses switch inputs and outputs.
2. Determining If a Function Has an Inverse
Use the horizontal line test to determine whether the function is one-to-one.
3. Identifying Inverse Behavior on Graphs
Recognize that inverse functions are reflections across the line \(y = x\).
4. Solving for the Inverse Algebraically (in Later Lessons)
Switch \(x\) and \(y\) and solve for the new \(y\).
Strategies
- When you see \(f(a) = b\), immediately know that \(f^{-1}(b) = a\).
- Use the horizontal line test to confirm one-to-one behavior.
- Remember that inverse functions swap inputs and outputs.
- Always distinguish between inverse functions and reciprocals—they are not the same.
- Think of inverses as undoing each step of the original function.
Worked Examples
Example 1 — Evaluating an Inverse
If:
\[ f(3) = 7, \]
then inverses reverse the mapping:
\[ f^{-1}(7) = 3. \]
Example 2 — Does a Function Have an Inverse?
A function has an inverse only if it passes the horizontal line test.
If a horizontal line touches the graph more than once, the function is not one-to-one and does not have an inverse function.
For example, \(f(x) = x^2\) does not have an inverse on all real numbers because many horizontal lines touch the parabola twice.
- Assuming every function has an inverse.
- Forgetting that inverses swap inputs and outputs.
- Confusing the reciprocal \(\dfrac{1}{f(x)}\) with the inverse function \(f^{-1}(x)\).
Practice Problems
- If \(f(2) = 9\), what is \(f^{-1}(9)\)?
- Does \(x^2\) have an inverse on all real numbers?
- What line is used when reflecting a function to get its inverse?
- If a function passes the horizontal line test, what does that tell you?
- If \(f(x) = x + 5\), find \(f^{-1}(x)\).
1.
\(f(2) = 9 \Rightarrow f^{-1}(9) = 2\)
2.
No. The function \(x^2\) fails the horizontal line test on all real numbers.
3.
The line is:
\[ y = x \]
4.
It tells you the function is one-to-one and has an inverse function.
5.
Let:
\[ y = x + 5 \]
Switch \(x\) and \(y\):
\[ x = y + 5 \]
Solve for \(y\):
\[ y = x - 5 \]
So:
\[ f^{-1}(x) = x - 5 \]
Summary
- An inverse function reverses the original function’s input-output relationship.
- Inverses satisfy \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
- A function must be one-to-one to have an inverse.
- Use the horizontal line test to check whether a graph is one-to-one.
- Graphs of inverses are reflections across the line \(y = x\).
- Inverse does not mean reciprocal.
- If you know \(f(a) = b\), then \(f^{-1}(b) = a\).
- Use the horizontal line test before trying to find an inverse.
- Reflect across \(y = x\) to visualize inverses.
- When solving for \(f^{-1}(x)\) algebraically: switch \(x\) and \(y\), then solve.