Inverse Function - Grade 11 General Mathematics

Introduction

This tutorial will guide you through the process of finding the inverse of a function, specifically focusing on linear functions as demonstrated in the Grade 11 General Mathematics video by Teacher Gon. Understanding inverse functions is essential in algebra and has applications in various fields, including engineering and computer science.

Step 1: Replace the Function with y

• Start by rewriting the function ( f(x) ) as ( y ).
• For example, if your function is ( f(x) = 3x + 4 ), you rewrite it as: [ y = 3x + 4 ]

Step 2: Interchange x and y

• Swap the variables ( x ) and ( y ).
• Using the previous example, the equation becomes: [ x = 3y + 4 ]

Step 3: Solve for y

• Rearrange the equation to isolate ( y ). Follow these steps:
  • Subtract 4 from both sides: [ x - 4 = 3y ]
  • Divide both sides by 3: [ y = \frac{x - 4}{3} ]

Step 4: Replace y with the Inverse Function Notation

• Replace ( y ) with the inverse function notation ( f^{-1}(x) ): [ f^{-1}(x) = \frac{x - 4}{3} ]

Step 5: Verify the Inverse Function

• To ensure the inverse is correct, check if ( f(f^{-1}(x)) = x ).
• Substitute ( f^{-1}(x) ) into the original function: [ f\left(f^{-1}(x)\right) = f\left(\frac{x - 4}{3}\right) = 3\left(\frac{x - 4}{3}\right) + 4 ] Simplifying this gives: [ x - 4 + 4 = x ]
• This confirms that the inverse function is correct.

Example 2: Finding the Inverse of a Rational Function

• Consider the function ( f(x) = \frac{x - 2}{2x + 3} ).

Step 1: Replace the Function with y

• Rewrite the function: [ y = \frac{x - 2}{2x + 3} ]

Step 2: Interchange x and y

• Swap the variables: [ x = \frac{y - 2}{2y + 3} ]

Step 3: Solve for y

• Cross-multiply: [ x(2y + 3) = y - 2 ]
• Distributing gives: [ 2xy + 3x = y - 2 ]
• Rearranging terms to isolate ( y ) leads to: [ 2xy - y = -3x - 2 ]
• Factor out ( y ): [ y(2x - 1) = -3x - 2 ]
• Solve for ( y ): [ y = \frac{-3x - 2}{2x - 1} ]

Step 4: Replace y with Inverse Function Notation

• Write the inverse function: [ f^{-1}(x) = \frac{-3x - 2}{2x - 1} ]

Step 5: Verify the Inverse Function

• Check if ( f(f^{-1}(x)) = x ):
  • Substitute ( f^{-1}(x) ) into the original function and simplify to confirm it equals ( x ).

Conclusion

Finding the inverse of a function involves a systematic approach of rewriting, interchanging, solving, and verifying. Practice with both linear and rational functions enhances your understanding of this important algebraic concept. Next, try finding inverses of more complex functions or explore their applications in real-world scenarios.