Overview
This lecture explains how to find the domain of different types of functions, using interval notation and logical reasoning for polynomials, rational functions, and functions involving square roots.
Domains of Basic Functions
- The domain of a linear function (e.g., 2x - 7) is all real numbers, written as (-∞, ∞).
- The domain of a quadratic or any polynomial (e.g., x² + 3x - 5, 2x³ - 5x² + 7x - 3) is also all real numbers, (-∞, ∞).
Rational Functions
- For rational functions (fractions), the domain excludes values that make the denominator zero.
- Example: For 5/(x - 2), set x - 2 ≠0, so x ≠2; domain: (-∞, 2) ∪ (2, ∞).
- For 3x - 8/(x² - 9x + 20), factor denominator to (x - 4)(x - 5) ≠0, so x ≠4, 5; domain: (-∞, 4) ∪ (4, 5) ∪ (5, ∞).
- If the denominator never equals zero (e.g., x² + 4), the domain is (-∞, ∞).
Functions with Square Roots
- For even-indexed roots, set the expression inside the root ≥ 0.
- Example: √(x - 4), set x - 4 ≥ 0 ⇒ x ≥ 4; domain: [4, ∞).
- For √(x² + 3x - 28), factor to (x - 4)(x + 7) ≥ 0; solution: x ≤ -7 or x ≥ 4; domain: (-∞, -7] ∪ [4, ∞).
Rational Functions with Square Roots
- If the square root is in the denominator, set inside > 0 (can't be zero).
- Example: 1/√(x + 3), x + 3 > 0 ⇒ x > -3; domain: (-3, ∞).
- If the square root is in the numerator, set inside ≥ 0.
- For fractions with roots in both numerator and denominator, determine valid regions for both and find their intersection.
Key Terms & Definitions
- Domain — Set of all possible x-values for which a function is defined.
- Interval Notation — Way of writing domains using parentheses for exclusive and brackets for inclusive endpoints.
- Rational Function — Function that is the ratio of two polynomials.
- Even-Indexed Root — Root with an even number (like square root); expression inside must be non-negative.
Action Items / Next Steps
- Practice finding domains for mixed types of functions.
- Review factoring and interval notation.
- Try setting up number lines for more complex domain questions.