To find the inverse function $g(x)$ of the given function $f(x) = \frac{x}{3} - 5$, we follow these steps:

1. Replace $f(x)$ with $y$: $y = \frac{x}{3} - 5$

2. Switch $x$ and $y$:
   To find the inverse, swap $x$ and $y$: $x = \frac{y}{3} - 5$

3. Solve for $y$:
   Now, solve for $y$ in terms of $x$. $x + 5 = \frac{y}{3}$ Multiply both sides by 3: $3(x + 5) = y$ So, $y = 3(x + 5)$

4. Write the inverse function:
   Since $y = g(x)$, we have: $g(x) = 3(x + 5)$

Thus, the inverse function is: $g(x) = 3(x + 5)$

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the function $f(x) = \frac{x}{3} - 5$ and its inverse?
3. How would the graph of the function $f(x)$ and its inverse $g(x)$ look like?
4. What happens to the inverse if we modify $f(x)$ to $f(x) = \frac{x}{3} + 5$?
5. Can a function have more than one inverse? Why or why not?

Tip: The graph of a function and its inverse are symmetrical across the line $y = x$.