Finding the Domain of Functions

1. Linear Functions

• Example: 2x - 7
  • Domain: All real numbers
  • In interval notation: (-∞, ∞)

2. Polynomial Functions

• Example: x² + 3x - 5
  • Domain: All real numbers
  • Example: 2x³ - 5x² + 7x - 3
  • Domain: All real numbers

3. Rational Functions

• Example: 5/(x - 2)
  • Domain: All real numbers except where denominator is zero
  • Set denominator not equal to 0: x - 2 ≠ 0
  • Result: x ≠ 2
  • In interval notation: (-∞, 2) ∪ (2, ∞)

Example Problem 2: (3x - 8)/(x² - 9x + 20)

• Set denominator not equal to 0: x² - 9x + 20 ≠ 0
• Factor the quadratic: (x - 4)(x - 5) ≠ 0
  • x ≠ 4, x ≠ 5
• Domain: (-∞, 4) ∪ (4, 5) ∪ (5, ∞)

Example Problem 3: (2x - 3)/(x² + 4)

• Set denominator not equal to 0: x² + 4 ≠ 0
  • Impossible since x² is always non-negative
• Domain: All real numbers (−∞, ∞)

4. Square Root Functions

• Example: √(x - 4)
  • Set inside greater than or equal to 0: x - 4 ≥ 0
  • Result: x ≥ 4
  • In interval notation: [4, ∞)

Example Problem 4: √(x² + 3x - 28)

• Set inside greater than or equal to 0: x² + 3x - 28 ≥ 0
• Factor: (x - 4)(x + 7) ≥ 0
  • x ≥ 4 or x ≤ -7
• Check intervals:
  • Domain: (-∞, -7] ∪ [4, ∞)

5. Square Roots in Denominator

• Example: 1/√(x + 3)
  • Inside must be > 0: x + 3 > 0
  • Result: x > -3
  • Domain: (-3, ∞)

6. Square Roots in Both Numerator and Denominator

• Example: √(x + 3)/√(x² - 16)
  • Numerator: x + 3 ≥ 0 -> x ≥ -3
  • Denominator: x² - 16 > 0; factors as (x + 4)(x - 4) > 0
  • Restrictions: x < -4 or x > 4

Intersection of Domains

• Combine results:
  • x ≥ -3, x < -4 or x > 4
• Valid region: Only x > 4
• Domain: (4, ∞)

Summary

• Understanding how to find domains for different types of functions: linear, polynomial, rational, and square root functions is crucial in algebra.
• Carefully analyze each component (numerator, denominator, and inside square roots) to determine all restrictions.