To find the inverse function of $f(x) = \frac{3}{5}x + 6$ in slope-intercept form, we will follow these steps:

Step 1: Rewrite $f(x)$ as $y$

$y = \frac{3}{5}x + 6$

Step 2: Swap $x$ and $y$ to find the inverse

$x = \frac{3}{5}y + 6$

Step 3: Solve for $y$

1. Subtract 6 from both sides: $x - 6 = \frac{3}{5}y$

2. Multiply both sides by $\frac{5}{3}$ to isolate $y$: $y = \frac{5}{3}(x - 6)$

3. Distribute $\frac{5}{3}$: $y = \frac{5}{3}x - 10$

Step 4: Rewrite as $f^{-1}(x)$

The inverse function is: $f^{-1}(x) = \frac{5}{3}x - 10$

Final Answer

The inverse function in slope-intercept form is: $f^{-1}(x) = \frac{5}{3}x - 10$

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. How do you verify if two functions are inverses of each other?
2. What does it mean for a function to be in slope-intercept form?
3. How would you find the inverse if the function was nonlinear, like $f(x) = x^2 + 6$?
4. What is the significance of the slope and y-intercept in the inverse function?
5. Can all functions have an inverse in the form $y = mx + b$?

Tip: Remember, when finding the inverse, swapping $x$ and $y$ is key. This reflects the function over the line $y = x$, which is a helpful way to visualize inverses.