To find the inverse of the function $f(x) = \frac{5}{2}x - 10$, we follow these steps:

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

$y = \frac{5}{2}x - 10$

Step 2: Swap $x$ and $y$

This is done because we are solving for the inverse function. $x = \frac{5}{2}y - 10$

Step 3: Solve for $y$

To isolate $y$, follow these steps:

1. Add 10 to both sides: $x + 10 = \frac{5}{2}y$
2. Multiply both sides by $\frac{2}{5}$ to cancel out the $\frac{5}{2}$ on the right: $y = \frac{2}{5}(x + 10)$

Step 4: Rewrite the inverse function

Now that $y$ is isolated, replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{2}{5}(x + 10)$

Final Answer

The inverse function in slope-intercept form is: $f^{-1}(x) = \frac{2}{5}x + 4$

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the domain and range of the original function?
2. How do you verify that two functions are inverses of each other?
3. Can a function always have an inverse? Why or why not?
4. What happens to the slope when finding the inverse of a linear function?
5. How does shifting the graph of a function affect its inverse?

Tip: When finding the inverse, remember to switch $x$ and $y$ and then solve for $y$.