Lecture Notes: Finding the Inverse of a Function

Introduction

• The lecture covers methods to find the inverse of a function.
• Key operations include replacing, switching, and isolating variables.

Basic Steps to Find the Inverse

1. Replace: Convert ( f(x) ) to ( y ).
2. Switch: Swap ( x ) with ( y ) in the equation.
3. Isolate: Solve the equation for ( y ).
4. Result: Express ( y ) in terms of ( x ) to get the inverse function.

Example 1: Linear Function

• Function: ( f(x) = 2x - 7 )
  • Replace: ( y = 2x - 7 )
  • Switch: ( x = 2y - 7 )
  • Solve: Add 7, divide by 2 to isolate ( y ).
  • Inverse: ( f^{-1}(x) = \frac{x + 7}{2} )

Example 2: Cubic Function

• Function: ( f(x) = x^3 + 8 )
  • Replace: ( y = x^3 + 8 )
  • Switch: ( x = y^3 + 8 )
  • Solve: Subtract 8, take cube root to isolate ( y ).
  • Inverse: ( f^{-1}(x) = \sqrt[3]{x - 8} )

Example 3: Square Root Function

• Function: ( f(x) = \sqrt{x + 2} - 5 )
  • Replace: ( y = \sqrt{x + 2} - 5 )
  • Switch: ( x = \sqrt{y + 2} - 5 )
  • Solve: Add 5, square both sides, simplify and isolate ( y ).
  • Inverse: ( f^{-1}(x) = x^2 + 10x + 23 ) or ( (x + 5)^2 - 2 )

Example 4: Cube Root Function

• Function: ( f(x) = \sqrt[3]{x + 4} - 2 )
  • Replace: ( y = \sqrt[3]{x + 4} - 2 )
  • Switch: ( x = \sqrt[3]{y + 4} - 2 )
  • Solve: Add 2, cube both sides, subtract 4.
  • Inverse: ( f^{-1}(x) = (x + 2)^3 - 4 )

Example 5: Rational Function

• Function: ( f(x) = \frac{3x - 7}{4x + 3} )
  • Replace: ( y = \frac{3x - 7}{4x + 3} )
  • Switch: ( x = \frac{3y - 7}{4y + 3} )
  • Solve: Cross-multiply, rearrange terms to isolate ( y ).
  • Inverse: ( f^{-1}(x) = \frac{3x + 7}{3 - 4x} )

Conclusion

• The inverse of a function reverses the original function's operations.
• Common techniques include switching variables, isolating terms, and using algebraic manipulations.