Algebraic and Graphical Verification of Inverses
The ultimate test of a relationship. If you go there and back again, do you end up exactly where you started?
Introduction
The ultimate test of a relationship. If you go there and back again, do you end up exactly where you started?
Past Knowledge
We know what composition is (Lesson 3.2) and what invertibility is (Lesson 3.3). Now we combine them to find the "undo" button for any function.
Today's Goal
We learn the Swap-and-Solve technique to find inverses algebraically, and the y=x Reflection property to verify them geometrically.
Future Success
Derivatives of inverse functions are related nicely. If has slope , then has slope at the reflected point. You need to be able to find and verify the inverse to use this theorem.
Key Concepts
The Verification Test
Two functions and are inverses if and only if they cancel each other out in BOTH directions.
Reflection over y = x
Since an inverse swaps X and Y inputs, the graph is literally flipped over the diagonal line .
Worked Examples
Example 1: The "Swap and Solve"
Find the inverse of .
Example 2: The Mirror
Verify graphically using and .
Example 3: Verifying by Composition
Verify is its own inverse (self-inverse).
We must show that .
My hypothesis was wrong! is NOT its own inverse. Let's find the real inverse. Swap and solve... .
Common Pitfalls
- The Power of -1:
is the INVERSE function.
is the RECIPROCAL .
These are completely different things. Notation is tricky! - Forgetting to verify BOTH ways:
Usually one way implies the other, but good mathematicians check both and to be sure domain issues don't crop up.
Real-Life Applications
Temperature Conversion
To go from Celsius to Fahrenheit: .
What if you have the Fahrenheit temperature and need Celsius? You need the inverse function!
Inverse: .
Using an inverse function is literally solving a formula for the "other variable."
Practice Quiz
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