Finding the Domain of Functions

Key Concepts

• Domain: Set of all possible x-values that can exist in a function.
• For simple polynomial functions (e.g., linear, quadratic), the domain is generally all real numbers.
• For rational functions, the domain excludes x-values that make the denominator zero.
• For functions involving square roots, ensure that the expression inside the square root is non-negative.

Linear and Polynomial Functions

• Linear Functions (e.g., 2x - 7): Domain is all real numbers.
  • Interval Notation: (-∞, ∞)
• Quadratic Functions (e.g., x² + 3x - 5): Domain is all real numbers.
  • Interval Notation: (-∞, ∞)
• Polynomial Functions (e.g., 2x³ - 5x² + 7x - 3): Domain is all real numbers.
  • Interval Notation: (-∞, ∞)

Rational Functions

• Example: 5 / (x - 2)

  • Domain excludes x = 2 (as it makes the denominator zero).
  • Interval Notation: (-∞, 2) ∪ (2, ∞)

• Example: (3x - 8) / (x² - 9x + 20)

  • Factor denominator to find x-values that make it zero: (x - 4)(x - 5) ≠ 0.
  • Exclude x = 4 and x = 5.
  • Interval Notation: (-∞, 4) ∪ (4, 5) ∪ (5, ∞)

Square Root Functions

• Example: √(x - 4)

  • Ensure x - 4 ≥ 0 -> x ≥ 4.
  • Interval Notation: [4, ∞)

• Example: √(x² + 3x - 28)

  • Set x² + 3x - 28 ≥ 0 and factor to find critical points.
  • Use test points to determine sign of regions.
  • Interval Notation: (-∞, -7] ∪ 4, ∞)

Fractions with Square Roots

• Square root in the denominator: Set inside > 0.

• Square root in the numerator: Set inside ≥ 0.

• Example: √(x + 3) / (√(x² - 16))

  • Square root numerator: x + 3 ≥ 0 -> x ≥ -3.
  • Square root denominator: x² - 16 > 0 -> factor as (x + 4)(x - 4).
  • Determine valid regions on a number line and find intersection.
  • Interval Notation: (4, ∞)

Conclusion

• Methods for finding domains vary depending on the function type.
• Use factoring and testing of intervals to determine which x-values are valid based on function constraints.