To find the inverse of the function $f(x) = 3x + 6$, follow these steps:

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

$y = 3x + 6$

Step 2: Swap $x$ and $y$

The inverse of a function is found by switching $x$ and $y$. This gives: $x = 3y + 6$

Step 3: Solve for $y$

Isolate $y$ to get the inverse function: $x - 6 = 3y$ $y = \frac{x - 6}{3}$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{x - 6}{3}$

Let me know if you would like more details or if you have any questions!

Here are 5 related questions to further explore:

1. What is the domain and range of $f(x) = 3x + 6$?
2. How do you verify that two functions are inverses of each other?
3. How does the graph of a function compare to the graph of its inverse?
4. What happens to the slope when you find the inverse of a linear function?
5. Can every function have an inverse?

Tip: Always check that a function is one-to-one before finding its inverse, as only one-to-one functions have inverses.