Finding the Domain of Functions

Linear Functions

• Example: ( f(x) = 2x - 7 )
• The domain of linear functions is all real numbers ((\mathbb{R})).
• In interval notation: ((-\infty, +\infty)).

Polynomial Functions

• Includes quadratic functions (e.g., (x^2 + 3x - 5)) and other polynomial forms (e.g., (2x^3 - 5x^2 + 7x - 3)).
• The domain is all real numbers ((\mathbb{R})).

Rational Functions

• Function has the form (\frac{p(x)}{q(x)}).
• Domain: All real numbers except where the denominator is zero.

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

• Set denominator not equal to zero: (x - 2 \neq 0)
• Domain: ((-\infty, 2) \cup (2, +\infty))

Example: (\frac{3x - 8}{x^2 - 9x + 20})

• Factor denominator: ((x - 4)(x - 5) \neq 0)
• Domain: ((-\infty, 4) \cup (4, 5) \cup (5, +\infty))

Square Root Functions

• Set the expression inside the square root greater than or equal to zero.

Example: (\sqrt{x - 4})

• Solve (x - 4 \geq 0) → (x \geq 4)
• Domain: ([4, +\infty))

Example: (\sqrt{x^2 + 3x - 28})

• Factor: ((x - 4)(x + 7))
• Solve: (x \leq -7) or (x \geq 4)
• Domain: ((-\infty, -7] \cup [4, +\infty))

Functions with Square Roots in Denominator

• The expression inside the square root must be greater than zero.
• Example: (\frac{1}{\sqrt{x + 3}}) → (x + 3 > 0) → (x > -3)
• Domain: ((-3, +\infty))

Fractions with Square Roots in Numerator and Denominator

• Set the numerator expression (\geq 0) and the denominator expression (> 0).
• Example: (\frac{\sqrt{x + 3}}{\sqrt{x^2 - 16}})
  • Numerator: (x + 3 \geq 0) → (x \geq -3)
  • Denominator: (x^2 - 16 > 0)
  • Factor: ((x + 4)(x - 4))
  • Check signs between intervals.
  • Domain: ((4, +\infty))

Summary

• Linear & Polynomial functions have a domain of all real numbers.
• Rational functions’ domain excludes values that make the denominator zero.
• Square root functions' domain is determined by setting expressions under the root (\geq 0).
• Complex functions with square roots in the denominator have specific restrictions to ensure no division by zero.