To find the inverse of the function $f(x) = \frac{2x - 3}{x - 1}$, follow these steps:

Step 1: Set $f(x) = y$

We begin by writing the equation as: $y = \frac{2x - 3}{x - 1}$

Step 2: Swap $x$ and $y$

To find the inverse, we switch $x$ and $y$ in the equation: $x = \frac{2y - 3}{y - 1}$

Step 3: Solve for $y$

Now, solve this equation for $y$.

1. Multiply both sides by $y - 1$ to eliminate the denominator: $x(y - 1) = 2y - 3$ $xy - x = 2y - 3$

2. Rearrange the equation to get all terms involving $y$ on one side: $xy - 2y = x - 3$

3. Factor out $y$ from the left side: $y(x - 2) = x - 3$

4. Solve for $y$: $y = \frac{x - 3}{x - 2}$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{x - 3}{x - 2}$

Would you like more details or clarifications?

Here are 5 related questions to explore:

1. What are the domain and range of the original function $f(x)$?
2. How do we verify that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$?
3. What is the domain and range of the inverse function $f^{-1}(x)$?
4. How does the vertical asymptote of $f(x)$ affect its inverse?
5. What happens to the graph of a function when you take its inverse?

Tip: Always check if the inverse function exists by confirming that the original function is one-to-one (i.e., passes the horizontal line test).