Lesson 3.3

Identifying Invertible Functions

Can you reverse time? Can you undo a mathematical operation? Only if the function is "One-to-One."

Introduction

Prerequisite Connection: In Lesson 1.1, we used the Vertical Line Test to see if a relation was a function. Now we are asking a deeper question: Is it a reversible function?

Today's Increment: We introduce the Horizontal Line Test. If a horizontal line hits the graph more than once, you cannot reverse it (because you wouldn't know which x-value to go back to).

Why This Matters for Calculus: Inverse Trigonometric Functions (arcsin, arccos) are crucial for integrating expression like . To define them, we must first understand invertibility.

Explanation of Key Concepts

The Horizontal Line Test (HLT)

One-to-One (Invertible)

Every y-value comes from EXACTLY one x-value. Any horizontal line intersects at most ONCE.

Example: Lines, Cubics

Many-to-One (Not Invertible)

A y-value is shared by multiple x-values. A horizontal line typically hits TWICE or more.

Example: Parabolas

Worked Examples

Level: Basic

Example 1: The Cubic Function

Is invertible?

Conclusion

YES. No matter where you draw a horizontal line, it hits the blue curve exactly once. The function is One-to-One.

Level: Intermediate

Example 2: The Absolute Value

Is invertible?

Analysis

The line

hits the graph at both

and

Conclusion

NO. If I told you the output was 2, you wouldn't know if the input was -2 or 2. Ambiguity means it's not invertible.

Level: Advanced (Calculus Prep)

Example 3: Algebraically Proving Invertibility

Use algebra to check .

Assume . If we can prove , it is One-to-One.

3a - 5 = 3b - 5

Add 5 to both sides...

3a = 3b

Divide by 3...

a = b

Since we arrived at

with no ambiguity (like

), the function is invertible.

Common Pitfalls

Confusing Tests:
Vertical Line Test: Is it a function?
Horizontal Line Test: Is it an INVERTIBLE function?
Forgetting the "Zero Slope" sections:
A constant function is a function. But it fails the HLT miserably (the horizontal line coincides with the graph infinitely many times). You cannot invert a flat line.

Real-World Application

Cryptography: One-to-One is Mandatory

When you send a credit card number over the internet, it gets encrypted by a mathematical function.

This function MUST be One-to-One. Why? Because the bank needs to distinguish your credit card number from everyone else's. If two different people's numbers mapped to the same encrypted code (failing the HLT), the bank couldn't tell who made the purchase!