Lesson 4.3
Intro to Inverses
An inverse function "undoes" what the original function did — like hitting the rewind button. If turns 2 into 5, then turns 5 back into 2.
Introduction
Think of a function as a process: multiply by 3, then add 5. The inverse reverses each step in backwards order: subtract 5, then divide by 3. This lesson builds the concept — Lesson 4.4 handles the algebra.
Past Knowledge
Composition of functions (4.2) — the verification tool for inverses.
Today's Goal
Understand what an inverse is, identify one-to-one functions, and use the Horizontal Line Test.
Future Success
Lesson 4.4 teaches you to find inverses algebraically by switching x and y.
Key Concepts
What Is an Inverse?
Definition
is the inverse of if:
Graphically, and are reflections over the line .
One-to-One Functions
Only one-to-one functions have inverses.
One-to-one: each output comes from exactly one input (no y-value repeats).
Horizontal Line Test (HLT)
If any horizontal line hits the graph more than once, the function is NOT one-to-one → no inverse.
Worked Examples
Example 1: Inverse from a Table
BasicGiven the table for , find .
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| f(x) | 5 | 8 | 11 | 14 |
Swap each (x, y) pair
| x | 5 | 8 | 11 | 14 |
|---|---|---|---|---|
| f⁻¹(x) | 1 | 2 | 3 | 4 |
To find the inverse table, swap inputs and outputs!
Example 2: Graphical Reflection
IntermediateVerify that and are reflections over .
The blue and green lines are mirror images across
Example 3: Horizontal Line Test
Key SkillDoes have an inverse?
The line hits the parabola at AND
Two inputs give the same output → NOT one-to-one → no inverse (unless we restrict the domain).
fails the HLT → no inverse function
Common Pitfalls
The notation is the inverse function, NOT "1 over f." The −1 is not an exponent!
Assuming Every Function Has an Inverse
Only one-to-one functions have inverses. Always check with the Horizontal Line Test first.
Real-Life Applications
Celsius-to-Fahrenheit conversion has an inverse . Every time you convert temperature the other direction, you're using an inverse function.
Practice Quiz
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