The problem asks to find the inverse of the function $f(x) = \frac{3x-1}{2x-6}$.

1. Problem Description

The problem asks to find the inverse of the function $f(x) = \frac{3x-1}{2x-6}$.

2. Solution Steps

Let $y = f(x)$. We have $y = \frac{3x-1}{2x-6}$. To find the inverse, we need to solve for $x$ in terms of $y$.

$y(2x-6) = 3x-1$

$2xy - 6y = 3x - 1$

$2xy - 3x = 6y - 1$

$x(2y - 3) = 6y - 1$

$x = \frac{6y - 1}{2y - 3}$

Now, we replace $x$ with $f^{-1}(y)$. So, we have $f^{-1}(y) = \frac{6y-1}{2y-3}$. Finally, we replace $y$ with $x$ to express the inverse function in terms of $x$.

$f^{-1}(x) = \frac{6x-1}{2x-3}$

3. Final Answer

$f^{-1}(x) = \frac{6x-1}{2x-3}$