We are given the function:

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

We need to find the inverse function, $f^{-1}(x)$.

Steps to find the inverse function:

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

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

2. Swap $x$ and $y$ to begin solving for $y$:

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

3. Solve for $y$:

Multiply both sides by $3y - 2$ to get rid of the denominator:

$x(3y - 2) = y - 1$

Expand the left-hand side:

$3xy - 2x = y - 1$

Move all terms involving $y$ to one side:

$3xy - y = 2x - 1$

Factor out $y$ on the left:

$y(3x - 1) = 2x - 1$

Now, solve for $y$:

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

Thus, the inverse function is:

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

Final Answer:

The correct choice is C.

Would you like more details on any step? Here are some related questions to expand understanding:

1. How do you verify if two functions are inverses of each other?
2. Can every function have an inverse? Why or why not?
3. What are the restrictions on the domain and range of $f(x)$ and $f^{-1}(x)$?
4. How does the vertical line test help in determining if a function is invertible?
5. What happens to the graph of a function when you take its inverse?

Tip: When finding the inverse of a rational function, be careful with handling the algebra when swapping and solving for the new variable.