To find the inverse of the function $f(x) = \frac{3x - 1}{x + 2}$, follow these steps:

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

$y = \frac{3x - 1}{x + 2}$

Step 2: Swap $x$ and $y$ to begin finding the inverse.

$x = \frac{3y - 1}{y + 2}$

Step 3: Solve for $y$.

First, eliminate the fraction by multiplying both sides by $y + 2$: $x(y + 2) = 3y - 1$ Distribute $x$ on the left side: $xy + 2x = 3y - 1$ Now, gather all terms involving $y$ on one side of the equation: $xy - 3y = -1 - 2x$ Factor out $y$ on the left side: $y(x - 3) = -2x - 1$ Finally, solve for $y$: $y = \frac{-2x - 1}{x - 3}$

Step 4: Write the inverse function.

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

So, the inverse of the function is $f^{-1}(x) = \frac{-2x - 1}{x - 3}$.

Would you like more details or have any questions?

Here are some related questions:

1. How do you verify if the inverse function is correct?
2. What are the domain and range of the inverse function?
3. How do you find the derivative of the inverse function?
4. Can the inverse function have any restrictions?
5. How do you determine if a function has an inverse?

Tip: When finding the inverse of a function, always remember to swap $x$ and $y$ and then solve for $y$. This ensures that you correctly derive the inverse function.