Domain and Range of Rational Functions

Introduction

• Today's topic: Finding the domain and range of rational functions.
• Example functions:
  • f(x) = 2/(x + 1)
  • g(x) = (x - 2)/(x + 2)
  • h(x) = (x² - 3x - 4)/(x + 1)

Definitions

• Domain: Set of all possible x values.
• Range: Set of all possible y values.

Example 1: f(x) = 2/(x + 1)

Finding the Domain

• Denominator: x + 1
• Set denominator ≠ 0:
  • x + 1 ≠ 0
  • x ≠ -1
• Domain:
  • Set of x such that x ∈ ℝ, x ≠ -1.

Finding the Range

• Replace f(x) with y:
  • y = 2/(x + 1)
• Cross-multiply to solve for x:
  • y(x + 1) = 2
  • x + 1 = 2/y
  • x = 2/y - 1
• Restriction: y ≠ 0 (denominator cannot be zero)
• Range:
  • Set of y such that y ∈ ℝ, y ≠ 0.

Example 2: g(x) = (x - 2)/(x + 2)

Finding the Domain

• Denominator: x + 2
• Set denominator ≠ 0:
  • x + 2 ≠ 0
  • x ≠ -2
• Domain:
  • Set of x such that x ∈ ℝ, x ≠ -2.

Finding the Range

• Replace g(x) with y:
  • y = (x - 2)/(x + 2)
• Cross-multiply:
  • y(x + 2) = x - 2
  • xy + 2y = x - 2
  • xy - x = -2y - 2
  • x(y - 1) = -2y - 2
  • x = (-2y - 2)/(y - 1)
• Restriction: y ≠ 1 (denominator cannot be zero)
• Range:
  • Set of y such that y ∈ ℝ, y ≠ 1.

Example 3: h(x) = (x² - 3x - 4)/(x + 1)

Finding the Domain

• Denominator: x + 1
• Set denominator ≠ 0:
  • x + 1 ≠ 0
  • x ≠ -1
• Domain:
  • Set of x such that x ∈ ℝ, x ≠ -1.

Finding the Range

• Replace h(x) with y:
  • y = (x² - 3x - 4)/(x + 1)
• Factor the numerator:
  • (x - 4)(x + 1)
• Cancel common factor (x + 1):
  • y = x - 4
• Restricted value: if x = -1, then y = -5
• Range:
  • Set of y such that y ∈ ℝ, y ≠ -5.

Conclusion

• Summary of domains and ranges for each function:
  • f(x): Domain: x ≠ -1, Range: y ≠ 0
  • g(x): Domain: x ≠ -2, Range: y ≠ 1
  • h(x): Domain: x ≠ -1, Range: y ≠ -5
• Questions or clarifications can be left in the comments.