Lesson 6.2

Domain & Excluded Values

The "Do Not Enter" signs of the mathematical world. Learning where functions break, crash, and explode.

1

Introduction

Prerequisite Connection: We know that and . But what is ? It is undefined. The calculator says ERROR.

Today's Increment: For rational functions , we must protect the function from ever dividing by zero. The "Domain" is simply the list of all safe inputs.

Why This Matters: In engineering, an excluded value often represents a physical breaking point—infinite resonance, short circuits, or structural failure. We calculate domain to know the safety limits.

2

Explanation of Key Concepts

The "Denominator Defense"

Denominator

To find the domain, ignore the numerator completely. Set the bottom equal to zero, solve for "bad inputs", and then throw them out.

Set Notation:

(All real numbers except 5)
Interval Notation:

(Skip over 5)
3

Worked Examples

Level: Basic

Example 1: Single Restriction

Find the domain of .

Step 1: Set Denominator ≠ 0
Step 2: Solve

Final Answer (Interval)
Level: Intermediate

Example 2: Multiple Restrictions

Find the domain of .

Factoring the denominator reveals hidden dangers.

Two bad values: and .
Domain:
Level: Advanced

Example 3: Double Trouble

Find the domain of .

We have two rules clashing here:
  1. Square Root Rule: Insides must be .
  2. Denominator Rule: Bottom must be .
Combined Rule: "Strictly Greater Than Zero"
Solve: .
Final Answer
(Notice the soft bracket!)
4

Common Pitfalls

  • Looking at the Numerator:

    If , THEN the top matters. But for pure polynomials like , the top accepts ANY number. Don't restrict the top unless there is a square root involved!

  • Mixing up U and Intersection:

    We use (Union) to say "this zone OR that zone are both fine."

5

Real-World Application

Electric Fields

Coulomb's Law states that Force is proportional to .

If the distance becomes 0, the force becomes infinite. In the real world, this means you can't push two electrons into the exact same point in space. The mathematics predicts a "Singularity"—a place where the laws of physics break down because the domain is restricted!

6

Practice Quiz

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