Lecture Notes: Finding the Domain of a Function

Key Concepts

• Domain of a Function: Set of all possible x-values that can exist in a function.

Linear and Polynomial Functions

• Linear Functions: Example - f(x) = 2x - 7

  • Domain: All real numbers ( (-\infty, \infty) ).

• Polynomial Functions: Example - f(x) = 2x³ - 5x² + 7x - 3

  • Domain: All real numbers ( (-\infty, \infty) ).

Rational Functions

• Rational Function Example: ( \frac{5}{x-2} )

  • Domain: Exclude values that make the denominator zero.
  • For ( x-2 \neq 0 ), x cannot be 2.
  • Interval Notation: ( (-\infty, 2) \cup (2, \infty) ).

• Complex Rational Function Example: ( \frac{3x - 8}{x² - 9x + 20} )

  • Set denominator ( x² - 9x + 20 \neq 0 ) and factor: ((x-4)(x-5)).
  • Exclude x = 4 and x = 5.
  • Interval Notation: ((-\infty, 4) \cup (4, 5) \cup (5, \infty)).

Square Root Functions

• Square Root in Numerator Example: ( \sqrt{x-4} )

  • x - 4 ≥ 0, so x ≥ 4.
  • Interval Notation: ([4, \infty)).

• Square Root in Denominator Example: ( \frac{1}{\sqrt{x² - 16}} )

  • x² - 16 > 0, factoring gives ((x+4)(x-4)).
  • Domain: x < -4 or x > 4
  • Interval Notation: ((-\infty, -4) \cup (4, \infty)).

Combined Square Root in Numerator and Denominator

• Example Problem: Numerator: ( \sqrt{x+3} ), Denominator: ( \sqrt{x² - 16} )
  • Numerator sets x ≥ -3.
  • Denominator: x² - 16 > 0 (x < -4 or x > 4).
  • Combined: x > 4 is the region where both conditions are true.
  • Interval Notation: ((4, \infty)).

Summary

• Linear/Polynomial Functions: Domain is all real numbers.
• Rational Functions: Denominator not equal to zero.
• Square Root Functions:
  • Numerator: set expression ≥ 0.
  • Denominator: set expression > 0.
• Fraction with Square Roots: Analyze numerator and denominator separately and find intersecting domain region.