The given function is:

$f(x) = \frac{5}{2}x - 5$

Steps to find the inverse of this function:

1. Replace $f(x)$ with $y$: $y = \frac{5}{2}x - 5$

2. Swap $x$ and $y$ to begin finding the inverse: $x = \frac{5}{2}y - 5$

3. Solve for $y$ (the inverse function):

   • Add 5 to both sides: $x + 5 = \frac{5}{2}y$
   • Multiply both sides by $\frac{2}{5}$ to isolate $y$: $y = \frac{2}{5}(x + 5)$

4. Rewrite the result: $f^{-1}(x) = \frac{2}{5}(x + 5)$

This is the inverse function in slope-intercept form.

Would you like more details on any of these steps? Here are some follow-up questions that can expand your understanding:

1. How do you verify if two functions are inverses of each other?
2. What is the slope of the inverse function, and how does it relate to the original?
3. Can every function be inverted? Why or why not?
4. How does transforming the function affect its graph?
5. What are real-life applications of finding inverse functions?

Tip: The graph of a function and its inverse are symmetric with respect to the line $y = x$.