Verified Answer :

  • Replace \(f(x)\) with \(y\), swap \(x\) and \(y\), then solve for \(y\) to find the inverse function.
  • The inverse of \(f(x) = 5x\) is \(f^{-1}(x) = \frac{1}{5}x\).
  • The inverse of \(f(x) = \frac{-6 - 3x}{2}\) is \(f^{-1}(x) = -\frac{2}{3}x - 2\).
  • Graphically, each pair of functions is symmetric with respect to the line \(y = x\).
  • Final Answer: \(f_1^{-1}(x) = \frac{1}{5}x\), \(f_2^{-1}(x) = -\frac{2}{3}x - 2\). \(\boxed{f_1^{-1}(x) = \frac{x}{5}, f_2^{-1}(x) = -\frac{2}{3}x - 2}\).

Explanation

  1. Analysis and Procedure for Finding Inverses

To find the inverse of a function \(f(x)\), we follow a standard procedure: replace \(f(x)\) with \(y\), swap the variables \(x\) and \(y\), and solve the resulting equation for the new \(y\). The resulting expression for \(y\) is the inverse function \(f^{-1}(x)\). Geometrically, the graph of an inverse function is the reflection of the original function's graph across the line \(y = x\). We will apply this to the two given functions.

  1. Finding the Inverse of \(f(x) = 5x\) and Graphing Procedure

For the function \(f(x) = 5x\):

  1. Write the function as \(y = 5x\).
  2. Swap the variables \(x\) and \(y\), giving \(x = 5y\).
  3. Solve for \(y\) by dividing both sides by 5: \(y = \frac{x}{5}\).

Thus, the inverse function is \(f^{-1}(x) = \frac{1}{5}x\).

To graph this, we choose two points on the original function, such as \((0, 0)\) and \((1, 5)\). For the inverse function, we plot the corresponding reflected points \((0, 0)\) and \((5, 1)\). Plotting these lines will reveal they are symmetric with respect to the line \(y = x\).

  1. Finding the Inverse of \(f(x) = \frac{-6 - 3x}{2}\) and Graphing Procedure

For the function \(f(x) = \frac{-6 - 3x}{2}\):

  1. Write the function as \(y = \frac{-6 - 3x}{2}\).
  2. Swap the variables \(x\) and \(y\), giving \(x = \frac{-6 - 3y}{2}\).
  3. Multiply both sides by 2 to clear the fraction: \(2x = -6 - 3y\).
  4. Rearrange to isolate the term with \(y\): \(3y = -6 - 2x\).
  5. Divide by 3: \(y = \frac{-6 - 2x}{3} = -2 - \frac{2}{3}x\).

Thus, the inverse function is \(f^{-1}(x) = -\frac{2}{3}x - 2\).

To graph this, we choose two points on the original function, such as \((0, -3)\) and \((-2, 0)\). For the inverse function, we plot the corresponding points \((-3, 0)\) and \((0, -2)\). These lines also demonstrate symmetry with respect to the line \(y = x\).

  1. Conclusion

The inverse of the first function is \(f^{-1}(x) = \frac{1}{5}x\), and the inverse of the second function is \(f^{-1}(x) = -\frac{2}{3}x - 2\). In both cases, the inverse is obtained by solving for the independent variable after swapping \(x\) and \(y\). Plotting these alongside their original functions confirms they are reflections across the line \(y = x\).

Examples

Understanding inverse functions is essential in fields like cryptography and data recovery. For example, if a function represents an encryption algorithm that scrambles a message, its inverse function acts as the decryption key that restores the original message. By graphing these functions, we visualize the balance and symmetry between the encryption and decryption processes, helping ensure that every input has exactly one valid output.

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