To find the inverse function of $f(x) = \frac{5}{3}x + 5$ in slope-intercept form, follow these steps:

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

2. Swap $x$ and $y$: $x = \frac{5}{3}y + 5$

3. Solve for $y$:

   • Subtract 5 from both sides: $x - 5 = \frac{5}{3}y$
   • Multiply both sides by $\frac{3}{5}$ to isolate $y$: $y = \frac{3}{5}(x - 5)$
   • Distribute $\frac{3}{5}$: $y = \frac{3}{5}x - 3$

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

So, the inverse function is: $f^{-1}(x) = \frac{3}{5}x - 3$

Would you like further details on any of the steps?

Here are some related questions that expand on this concept:

1. How do you verify if two functions are inverses of each other?
2. What are the characteristics of a function that has an inverse?
3. How does changing the slope or intercept of a function affect its inverse?
4. How do you find the inverse of a function with more complex terms, like $f(x) = ax^2 + bx + c$?
5. What is the significance of the line $y = x$ when analyzing inverse functions?

Tip: When finding inverses of linear functions, remember that the inverse will also be linear, just with the roles of $x$ and $y$ switched.