Lecture Notes: Inverse Functions and Graphs

Introduction to Inverse Functions

• Definition: An inverse function is a function that reverses the operations done by the original function.
• Example:
  • Original function: ( f(x) = 3x + 9 )
  • Inverse function: ( f^{-1}(x) = \frac{x - 9}{3} )

Steps to Find an Inverse Function

1. Replace ( f(x) ) with ( y ).
2. Switch ( x ) and ( y ).
3. Solve for ( y ).
   • Example: ( y = 3x + 9 )
     • Switch to ( x = 3y + 9 )
     • Solve: ( y = \frac{x - 9}{3} )

Examples

• Example 1: ( f(x) = x^2 - 4 )
  • Inverse: ( f^{-1}(x) = \pm \sqrt{x + 4} )
• Example 2: ( f(x) = \sqrt[3]{3x + 8} )
  • Inverse: ( f^{-1}(x) = \frac{x^3 - 8}{3} )

Verifying Inverse Functions

• Goal: Show ( f(g(x)) = x ) and ( g(f(x)) = x ).
• Example:
  • If ( f(x) = 3x + 9 ) and ( g(x) = \frac{x - 9}{3} ), then both are inverses because both compositions equal ( x ).

Graphical Representation of Inverse Functions

• Symmetry: Inverses are symmetric about the line ( y = x ).
• Example: ( f(x) = x^2 ), inverse ( f^{-1}(x) = \pm \sqrt{x} ).

Function Tests

• Vertical Line Test: Determines if a graph represents a function.
• Horizontal Line Test: Determines if a function is one-to-one, which indicates the inverse will be a function.

Determining if the Inverse is a Function

• If ( f(x) ) passes the horizontal line test (one-to-one), the inverse is a function.
• Example:
  • ( f(x) = x^2 ): Fails horizontal line test, so inverse is not a function.
  • Right side of ( f(x) = x^2 ) (( y = \sqrt{x} )): Passes horizontal test, so inverse is a function.

Conclusion

• The horizontal line test is crucial for determining if an inverse is a function.
• If ( f(x) ) fails the horizontal line test, the inverse function will not pass the vertical line test, thus not being a function.