To find the inverse function of $f(x) = -\frac{3}{2}x + 12$ in slope-intercept form, we can follow these steps:

1. Rewrite $f(x)$ as $y$: $y = -\frac{3}{2}x + 12$

2. Swap $x$ and $y$: This is a key step in finding the inverse, as it essentially reverses the roles of the input and output. $x = -\frac{3}{2}y + 12$

3. Solve for $y$:

   • First, isolate the term with $y$: $x - 12 = -\frac{3}{2}y$
   • Then, multiply both sides by $-\frac{2}{3}$ to solve for $y$: $y = -\frac{2}{3}(x - 12)$
   • Distribute $-\frac{2}{3}$: $y = -\frac{2}{3}x + 8$

4. Rewrite $y$ as $f^{-1}(x)$: $f^{-1}(x) = -\frac{2}{3}x + 8$

So, the inverse function in slope-intercept form is: $f^{-1}(x) = -\frac{2}{3}x + 8$

Would you like more details on any step or have further questions?

Here are some related questions:

1. How do we find the inverse of a function in general?
2. Why do we switch $x$ and $y$ when finding the inverse?
3. How can we verify that two functions are inverses of each other?
4. What happens if the function is not one-to-one? Can we still find an inverse?
5. How does the slope of the original function relate to the slope of its inverse?

Tip: To verify if two functions are inverses, compose them (i.e., $f(f^{-1}(x))$ and $f^{-1}(f(x))$) and see if the result is $x$ for both compositions.