To find the inverse of the given function $f(x) = \frac{x - 6}{3x - 4}$, we will follow these steps:

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

$y = \frac{x - 6}{3x - 4}$

Step 2: Swap $x$ and $y$.

$x = \frac{y - 6}{3y - 4}$

Step 3: Solve for $y$.

To isolate $y$, multiply both sides by $3y - 4$: $x(3y - 4) = y - 6$ Expanding this gives: $3xy - 4x = y - 6$

Step 4: Rearrange the equation.

Bring all terms involving $y$ to one side: $3xy - y = 4x - 6$

Factor out $y$: $y(3x - 1) = 4x - 6$

Step 5: Solve for $y$.

$y = \frac{4x - 6}{3x - 1}$

Step 6: Write the inverse function.

$f^{-1}(x) = \frac{4x - 6}{3x - 1}$

The inverse function of $f(x)$ is $f^{-1}(x) = \frac{4x - 6}{3x - 1}$.

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to deepen your understanding:

1. How do you verify if two functions are inverses of each other?
2. What is the domain of the function $f(x)$ and its inverse $f^{-1}(x)$?
3. How do you find the inverse of a quadratic function?
4. What happens to the graph of $f(x)$ when you find its inverse?
5. How do you determine if a function is one-to-one?

Tip: To find the inverse of a function, always start by swapping $x$ and $y$, then solve for $y$.