Finding the Domain of a Function

Linear Functions

• Example: 2x - 7
• Domain: All real numbers (ℝ)
• Interval Notation: (-∞, ∞)

Quadratic and Polynomial Functions

• Examples: x^2 + 3x - 5, 2x^3 - 5x^2 + 7x - 3
• Domain: All real numbers (ℝ)
• Interval Notation: (-∞, ∞)

Rational Functions

• Key Idea: Denominator cannot be zero.
• Example: 5/(x - 2)
  • Set denominator ≠ 0
  • x - 2 ≠ 0 → x ≠ 2
  • Interval Notation: (-∞, 2) ∪ (2, ∞)
• Example: (3x - 8)/(x² - 9x + 20)
  • Factor denominator: (x - 4)(x - 5) ≠ 0
  • x ≠ 4, x ≠ 5
  • Interval Notation: (-∞, 4) ∪ (4, 5) ∪ (5, ∞)

Rational Function with Always Positive Denominator

• Example: (2x - 3)/(x² + 4)
  • x² + 4 will never be 0
  • Domain: All real numbers (ℝ)
  • Interval Notation: (-∞, ∞)

Square Root Functions

• Key Idea: Expression inside square root must be ≥ 0 (if index is even)
• Example: √(x - 4)
  • x - 4 ≥ 0 → x ≥ 4
  • Interval Notation: [4, ∞)
• Example: √(x² + 3x - 28)
  • Factor: (x - 4)(x + 7) ≥ 0
  • Check regions on number line
  • Interval Notation: (-∞, -7] ∪ [4, ∞)

Fraction with Square Root in Denominator

• Key Idea: Expression inside square root > 0
• Example: √(x + 3)/(x² - 16)
  • Factor denominator: (x + 4)(x - 4) > 0
  • Perform sign test
  • Interval Notation: (4, ∞)

Fraction with Square Root in Numerator and Denominator

• Example: √(x + 3)/(√(x² - 16))
  • Numerator: x + 3 ≥ 0 → x ≥ -3
  • Denominator: x² - 16 > 0 → x < -4, x > 4
  • Intersection of valid regions
  • Interval Notation: (4, ∞)

Summary

• For linear, quadratic, and polynomial functions, the domain is all real numbers unless specified otherwise.
• For rational functions, exclude values that make the denominator zero.
• For square root functions, ensure the expression inside the square root is non-negative.
• Apply sign tests for multiple conditions in complex rational functions with square roots.