To find the inverse of the function $f(x) = \frac{5x - 6}{x - 1}$, we need to follow these steps:

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

$y = \frac{5x - 6}{x - 1}$

Step 2: Swap $x$ and $y$, then solve for $y$.

$x = \frac{5y - 6}{y - 1}$

Step 3: Multiply both sides by $y - 1$ to eliminate the denominator.

$x(y - 1) = 5y - 6$ $xy - x = 5y - 6$

Step 4: Group all terms with $y$ on one side.

$xy - 5y = x - 6$

Step 5: Factor out $y$ on the left-hand side.

$y(x - 5) = x - 6$

Step 6: Solve for $y$.

$y = \frac{x - 6}{x - 5}$

Thus, the inverse of $f(x)$ is: $f^{-1}(x) = \frac{x - 6}{x - 5}$

Would you like more details or have any questions?

Here are some related questions:

1. How do you verify if two functions are inverses of each other?
2. What are the key characteristics of inverse functions?
3. How do horizontal line tests help in finding inverses?
4. Can all rational functions have inverses?
5. What happens to domain and range when finding an inverse?

Tip: Always check if the original function passes the horizontal line test to ensure it has an inverse.