You have provided the function $f(x) = \frac{x - 1}{2x + 3}$ and are asking for its inverse, denoted as $f^{\text{inv}}(x)$.

To find the inverse of a function $f(x)$, we follow these steps:

1. Start with the equation: $y = \frac{x - 1}{2x + 3}$

2. Swap $x$ and $y$ to find the inverse: $x = \frac{y - 1}{2y + 3}$

3. Now, solve for $y$: Multiply both sides by $2y + 3$ to eliminate the denominator: $x(2y + 3) = y - 1$

4. Distribute $x$ on the left-hand side: $2xy + 3x = y - 1$

5. Move all terms involving $y$ to one side: $2xy - y = -1 - 3x$ Factor out $y$ on the left-hand side: $y(2x - 1) = -1 - 3x$

6. Solve for $y$: $y = \frac{-1 - 3x}{2x - 1}$

Thus, the inverse function is: $f^{\text{inv}}(x) = \frac{-1 - 3x}{2x - 1}$

Let me know if you'd like any additional explanation or steps broken down!

Here are 5 related questions:

1. How can you verify that two functions are inverses of each other?
2. What are the domain and range of the inverse function $f^{\text{inv}}(x)$?
3. How does finding the inverse change if the function is more complex?
4. Can every function have an inverse? What are the conditions?
5. How would you graph both the function and its inverse?

Tip: When finding an inverse, always ensure the original function is one-to-one (passes the horizontal line test). This guarantees that the inverse exists.