We are given the function $g(x) = \frac{3x-4}{11x}$ and we need to find its inverse function $g^{-1}(y)$.

1. Problem Description

We are given the function $g(x) = \frac{3x-4}{11x}$ and we need to find its inverse function $g^{-1}(y)$.

2. Solution Steps

To find the inverse function $g^{-1}(y)$, we first replace $g(x)$ with $y$:

$y = \frac{3x-4}{11x}$

Next, we swap $x$ and $y$:

$x = \frac{3y-4}{11y}$

Now, we solve for $y$ in terms of $x$:

$11xy = 3y - 4$

$11xy - 3y = -4$

$y(11x - 3) = -4$

$y = \frac{-4}{11x - 3}$

$y = \frac{4}{3 - 11x}$

Finally, we replace $x$ with $y$ to express the inverse function in terms of $y$:

$g^{-1}(y) = \frac{4}{3 - 11y}$

3. Final Answer

$g^{-1}(y) = \frac{4}{3-11y}$