Lesson 3.4

Restricting Domains for Invertibility

Sometimes you have to cut part of the graph away to save the rest. How to force a function to be One-to-One.

Introduction

Prerequisite Connection: We learned in Lesson 3.3 that parabolas FAIL the Horizontal Line Test. Does that mean we can never undo a square? Of course not! We have square roots, don't we?

Today's Increment: We resolve this paradox by Restricting the Domain. By deleting half the parabola, the remaining half passes the HLT, allowing us to define an inverse.

Why This Matters for Calculus: This is literally how , , and are defined. Without domain restriction, Trigonometry would be a dead end.

Explanation of Key Concepts

The "Cut" Strategy

To make a function One-to-One, we must remove parts of the domain until no two inputs share the same output.

Step 1: Identify the Turning Point

Find the vertex (for parabolas) or the peaks/valleys (for trig). This is usually where the function doubles back on itself.

Step 2: Keep ONE Side

By convention, we usually keep the positive side or the interval containing zero, but mathematically, either side works.

Worked Examples

Level: Basic

Example 1: Fixing x squared

Make invertible.

Choice

We keep

Result

The new function

passes the Horizontal Line Test. Its inverse is

Level: Intermediate

Example 2: Finding the Vertex

Restrict the domain of .

Analysis

The axis of symmetry is

. We must slice it here.

Answer

Restrict domain to

Level: Advanced

Example 3: Defining Arcsin

How do we restrict to define its inverse?

We pick the interval

. Why?

It covers all possible output values (-1 to 1).
It is one contiguous chunk near zero.
It passes the HLT.

Common Pitfalls

Forgetting the "Other Side":
Students assume is the ONLY way. Restricting to also works perfectly fine! It just creates a different inverse function (negative square root).
Cutting in the wrong spot:
If you cut a parabola at instead of the vertex , the piece from 0 to 1 will still cause HLT failure for the other side. You MUST cut exactly at the turning point.

Real-World Application

Physics: Projectile Motion

A ball thrown in the air follows a parabolic path: height .

Mathematically, this parabola exists for negative time (). But physically, the ball wasn't thrown yet. We restrict the domain to . This restriction makes the height function invertible, allowing us to ask "At what time was the ball at 50 feet?" without getting a negative answer.