We are given the function $f(x) = \frac{9}{5}x + 32$, which converts degrees Celsius $x$ to degrees Fahrenheit. The task is to find the inverse function $f^{-1}(x)$, which will convert degrees Fahrenheit back to degrees Celsius.

Step-by-step solution to find the inverse function:

1. Start with the given function:
   $y = \frac{9}{5}x + 32$ where $y$ represents Fahrenheit and $x$ represents Celsius.

2. Swap $x$ and $y$ to find the inverse:
   $x = \frac{9}{5}y + 32$

3. Solve for $y$ (the new function):
   Subtract 32 from both sides: $x - 32 = \frac{9}{5}y$ Multiply both sides by $\frac{5}{9}$ to isolate $y$: $y = \frac{5}{9}(x - 32)$

Thus, the inverse function is: $f^{-1}(x) = \frac{5}{9}(x - 32)$

Interpretation of the inverse function:

The inverse function $f^{-1}(x)$ represents the conversion from degrees Fahrenheit back to degrees Celsius. It takes a Fahrenheit temperature as input and returns the corresponding Celsius temperature.

Would you like any additional details or have any questions on this? Here are five questions that might help expand your understanding:

1. What are the steps to verify that $f^{-1}(x)$ is the correct inverse of $f(x)$?
2. Why do we swap $x$ and $y$ when finding the inverse function?
3. How would you use $f^{-1}(x)$ to convert 98°F to Celsius?
4. Can every function have an inverse? Why or why not?
5. What is the graphical interpretation of inverse functions?

Tip: When solving for the inverse function, always ensure that your final equation expresses the dependent variable in terms of the independent variable (i.e., solve for $y$ in terms of $x$).