Inverse Functions - Domain & range- With Fractions, Square Roots, & Graphs

Inverse Functions Introduction

• The inverse function is formed by switching the X and Y values of the original function.

• Example: If ( f(3) = 1 ), then ( f^{-1}(1) = 3 ).

• Inverse function notation: ( f^{-1}(x) ).

Finding the Inverse Algebraically

Example 1: Linear Function

• Given: ( f(x) = 3x - 8 )

  • Replace ( f(x) ) with ( Y ): ( Y = 3x - 8 ).

  • Switch X and Y: ( X = 3Y - 8 ).

  • Solve for Y:

    • Add 8 to both sides: ( X + 8 = 3Y ).

    • Divide by 3: ( Y = \frac{X + 8}{3} ).

  • Inverse function: ( f^{-1}(x) = \frac{x + 8}{3} ).

Example 2: Rational Function

• Given: ( f(x) = \frac{2x + 5}{3x - 1} )

  • Replace ( f(x) ) with ( Y ): ( Y = \frac{2x + 5}{3x - 1} ).

  • Switch X and Y: ( X = \frac{2Y + 5}{3Y - 1} ).

  • Cross-multiply: ( 1(2Y + 5) = X(3Y - 1) ).

  • Expand: ( 2Y + 5 = 3XY - X ).

  • Rearrange: ( 3XY - 2Y = 3X + 5 ).

  • Factor: ( Y(3X - 2) = 3X + 5 ).

  • Divide by ( 3X - 2 ): ( Y = \frac{3X + 5}{3X - 2} ).

Example 3: Square Root Function

• Given: ( f(x) = \sqrt{2x + 6} )

  • Replace with Y: ( Y = \sqrt{2x + 6} ).

  • Switch X and Y: ( X = \sqrt{2Y + 6} ).

  • Square both sides: ( X^2 = 2Y + 6 ).

  • Solve for Y: ( 2Y = X^2 - 6 ) → ( Y = \frac{X^2 - 6}{2} ).

  • Inverse function: ( f^{-1}(x) = \frac{x^2 - 6}{2} ).

Domain and Range Analysis

• To find the domain and range:

  • The domain of the original function becomes the range of the inverse.

  • The range of the original function becomes the domain of the inverse.

• Example: ( f(x) = \sqrt{2x + 6} )

  • Domain: ( x \geq 3 ) → Range: ( Y \geq 0 ).

  • Inverse: Domain: ( Y \geq 0 ) → Range: ( Y \geq 3 ).

Graphing Inverse Functions

• The graph of an inverse function has symmetry about the line ( y = x ).

• Example: Graph of a square root function shows restricted domain and range; inverse is part of a parabola.

  • For squared functions, only the right half is considered.

Additional Examples & Concepts

Example 4: Cube Root Function

• Given: ( f(x) = \sqrt[3]{x - 4} + 1 )

  • Replace, switch, and solve for Y to find the inverse: ( f^{-1}(x) = (x - 1)^3 + 4 ).

• Domain/Range: All real numbers due to the odd index.

Example 5: Exponential and Logarithmic Functions

• Given: ( f(x) = e^{3x + 1} - 5 )

  • Inverse function found by taking logarithm after switching X and Y: ( f^{-1}(x) = \frac{1}{3} \ln(x + 5) - \frac{1}{3} ).

• Domain and range are also inversely related:

  • Domain of ( e^x ): all real numbers → Range: ( Y > 0 ).

  • Domain of ( \ln(x) ): ( X > 0 ) → Range: all real numbers.

Proving Inverse Functions

• To prove two functions are inverses, verify:

  • ( f(g(x)) = x )

  • ( g(f(x)) = x )

• Example: For ( f(x) = \sqrt[3]{2x - 7} ) and ( g(x) = \frac{x^3 + 7}{2} ): both conditions confirm they are inverses.

Tests for Functions

• Vertical Line Test: To check if a relation is a function; it should intersect a vertical line at most once.

• Horizontal Line Test: To determine if a function has an inverse; a function must pass this test to ensure every Y has one corresponding X.

Summary of Function Properties

• Functions that are bijective (one-to-one) have inverses that are also functions.

• If a function fails the horizontal line test, its inverse may not pass the vertical line test.