Operations of Functions and Domain Issues

Introduction

• Focus on operations of functions and domain
• Adding, subtracting, multiplying, and dividing functions
• Importance of the domain in these operations
• Domain cannot be canceled out or ignored

Domain of Functions

• Domain is like the skeletons in the closet; can't get rid of them
• Domain carries its baggage with it
• When composing, adding, subtracting, multiplying, or dividing functions, domain issues persist
• May develop new domain problems through these operations

Steps to Handle the Domain

1. Find the domain of each function individually
   • Example: For fractions, look for denominators that do not equal zero
   • Denominator equaling zero makes the output undefined
   • Example problem: if x = 2/3, denominator is zero (undefined)
2. When adding functions, domains combine to form new domains but retain original issues
   • Common denominator keeps same domain problem
   • Example: (6x + 3) / (3x - 2)
3. Subtraction, domain issues persist similarly
   • Example: (2x + 3 - 4x) / (3x - 2)

Multiplication and Domain

• Multiply fractions: new denominators replicate original domain issues
• Factors should remain factored, especially for graphing
• Domain doesn't change; retains original restrictions
• Example: (2x + 3) / (3x - 2) * 4x / (3x - 2)

Division and Added Domain Issues

• Division may hide domain issues, but they still exist
• Complex fractions: Use reciprocal to find resultant functions
• Example: (2x + 3) / (3x - 2) ÷ (4x / 3x - 2)
  • Reciprocate and multiply: (2x + 3) / (3x - 2) * (3x - 2) / 4x
  • Result: domain restrictions remain the same
  • New issues can emerge but cannot ignore initial domain problems

Special Case: Square Roots

• Square roots need non-negative values
  • Example: sqrt(x + 3)
  • x + 3 ≥ 0 ⟹ x ≥ -3
• Combine domain restrictions of square roots and fractions
  • Example: avoid x = 0 in fractions

General rule:

• Find domain first and apply throughout the operations
• Cannot cancel out domain problems
• Write down domains at the start as a guide

Applications and Future Lectures

• Next topic: Graphing functions and the vertical line test
• Discuss even vs. odd functions
• Preview future exploration on the subject