To find the inverse of the given function $f(x) = \frac{4 - x}{3x}$, we need to follow these steps:

Step 1: Replace $f(x)$ with $y$

Let $y = \frac{4 - x}{3x}$.

Step 2: Swap $x$ and $y$

This is the key step in finding the inverse: $x = \frac{4 - y}{3y}.$

Step 3: Solve for $y$

Multiply through by $3y$ to eliminate the fraction: $x(3y) = 4 - y.$ $3xy = 4 - y.$

Rearrange terms to isolate $y$: $3xy + y = 4.$ $y(3x + 1) = 4.$

Solve for $y$: $y = \frac{4}{3x + 1}.$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{4}{3x + 1}.$

Let me know if you'd like additional explanation or verification!

Questions for deeper understanding:

1. What is the general procedure to find the inverse of a rational function?
2. How can we verify that two functions are inverses of each other?
3. Why do we swap $x$ and $y$ when finding the inverse?
4. What are the domain and range restrictions for $f(x)$ and $f^{-1}(x)$?
5. Could the function $f(x)$ have more than one inverse? Why or why not?

Tip: Always check for restrictions in the domain when finding inverses, as division by zero or other undefined points can impact the validity of the inverse.