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
Sometimes you have to cut part of the graph away to save the rest. How to force a function to be One-to-One.
Past Knowledge
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 Goal
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.
Future Success
This is literally how , , and are defined. Without domain restriction, Trigonometry would be a dead end.
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.
Find the vertex (for parabolas) or the peaks/valleys (for trig). This is usually where the function doubles back on itself.
By convention, we usually keep the positive side or the interval containing zero, but mathematically, either side works.
Worked Examples
Example 1: Fixing x squared
Make invertible.
Example 2: Finding the Vertex
Restrict the domain of .
Example 3: Defining Arcsin
How do we restrict to define its inverse?
- 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-Life Applications
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.
Practice Quiz
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