The given question asks for the inverse of a function $f(x-1) = \frac{x-1}{2x-1}, \text{ with } x \neq \frac{1}{2}$, and how to find $f^{-1}(2x-1)$.

Steps to find the inverse function:

1. Express the function $f(y) = \frac{y - 1}{2y - 1}$, where $y = x - 1$.

2. Solve for $y$ in terms of $f(y)$: $f(y) = \frac{y - 1}{2y - 1}$ Let $z = f(y)$, so: $z = \frac{y - 1}{2y - 1}$ Multiply both sides by $2y - 1$ to eliminate the denominator: $z(2y - 1) = y - 1$ Expanding both sides: $2zy - z = y - 1$ Rearranging to collect $y$-terms on one side: $2zy - y = z - 1$ Factor out $y$ on the left: $y(2z - 1) = z - 1$ Solving for $y$: $y = \frac{z - 1}{2z - 1}$

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

4. Apply the inverse function to $2x - 1$: $f^{-1}(2x - 1) = \frac{(2x - 1) - 1}{2(2x - 1) - 1}$ Simplify the numerator and the denominator: $f^{-1}(2x - 1) = \frac{2x - 2}{4x - 2 - 1} = \frac{2x - 2}{4x - 3}$ Factor out the 2 from the numerator: $f^{-1}(2x - 1) = \frac{2(x - 1)}{4x - 3}$

Thus, the correct answer is:

$\boxed{b. \frac{2x - 1}{4x - 3}, x \neq \frac{3}{4}}$

Let me know if you'd like any further details or have any questions!

Here are five related questions:

1. How do you calculate the inverse of a function in general?
2. What are the conditions for the existence of an inverse function?
3. Why do we check for domain restrictions in inverse functions?
4. How would you graph a function and its inverse?
5. How do inverse functions relate to real-world applications?

Tip: Always check the domain and range restrictions when finding inverse functions to ensure validity.