Finding the Domain of an Inverse Function

Overview

• Focus on how to find the domain of an inverse function.
• Understand concepts of domain and range.
• Examine various function types and restrictions.

General Concepts

• Domain: X values a function can have.
• Range: Y values a function can take.
• One-to-One Function: Must pass the horizontal line test to ensure an inverse exists.

Example 1: Linear Function

• Function: f(x) = 3x - 4
  • Graph: Y-intercept at -4; slope of 3.
  • Domain & Range: Both from (-∞, ∞).
• Inverse Function: Swap x and y, solve for y.
  • Result: Inverse function is (x + 4) / 3.
  • Domain of Inverse: Same as the range of the original, (-∞, ∞).

Example 2: Quadratic Function

• Function: f(x) = x² - 3
  • Graph: Parabola shifted 3 units down.
  • Horizontal Line Test: Does not pass without restriction.
• Restriction: Use only x ≥ 0.
  • Restricted Domain: [0, ∞)
  • Range: [-3, ∞)
• Inverse Function: x = √(x + 3)
  • Graph reflects about y = x.
  • Domain of Inverse: -3, ∞)

Example 3: Radical Function

• Function: f(x) = √(8 - 2x)
  • Domain: Solve 8 - 2x ≥ 0, resulting in (-∞, 4].
  • Range: [0, ∞) as y values start at 0.
• Inverse Function: y = 8 - x² / 2
  • Graph reflects about y = x.
  • Domain of Inverse: 0, ∞)

Example 4: Rational Function

• Function: f(x) = (3x + 2) / (5x - 4)
  • Domain: x ≠ 4/5 due to zero in denominator.
  • Range: y ≠ 3/5 (horizontal asymptote).
• Inverse Function: y = (4x + 2) / (5x - 3)
  • Domain of Inverse: x ≠ 3/5, reflecting range of original.

Key Takeaways

• To find the domain of an inverse function, determine the range of the original function.
• Ensure the original function is one-to-one by restricting domains if necessary.
• Analyze function types (linear, quadratic, radical, rational) to apply correct methods.