To find the inverse of the function $f(x) = -\frac{2}{3}x + 10$, follow these steps:

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

$y = -\frac{2}{3}x + 10$

Step 2: Swap $x$ and $y$:

$x = -\frac{2}{3}y + 10$

Step 3: Solve for $y$ (this will give the inverse function):

1. Subtract 10 from both sides: $x - 10 = -\frac{2}{3}y$
2. Multiply both sides by $-\frac{3}{2}$ to isolate $y$: $y = -\frac{3}{2}(x - 10)$
3. Distribute the multiplication: $y = -\frac{3}{2}x + 15$

Step 4: Rewrite the inverse function:

$f^{-1}(x) = -\frac{3}{2}x + 15$

Thus, the inverse function is: $f^{-1}(x) = -\frac{3}{2}x + 15$

Would you like more details on this process or have any questions?

Related Questions:

1. What is the geometric interpretation of an inverse function?
2. How do you verify that two functions are inverses of each other?
3. What happens to the domain and range when finding the inverse of a function?
4. How do you find the inverse of a quadratic function?
5. Can every linear function have an inverse?

Tip: When finding the inverse, remember that switching $x$ and $y$ reflects the graph over the line $y = x$.