The given function is:

$f(x) = \frac{2x - 1}{3x + 4}$

with the condition $x \neq -\frac{4}{3}$.

We are tasked with finding the inverse function $f^{-1}(x)$.

Steps to find the inverse function:

1. Replace $f(x)$ with $y$: $y = \frac{2x - 1}{3x + 4}$

2. Switch $x$ and $y$ to find the inverse: $x = \frac{2y - 1}{3y + 4}$

3. Solve for $y$ (the inverse function):

   • Multiply both sides by $3y + 4$: $x(3y + 4) = 2y - 1$
   • Expand: $3xy + 4x = 2y - 1$
   • Collect terms involving $y$ on one side: $3xy - 2y = -1 - 4x$
   • Factor out $y$ on the left-hand side: $y(3x - 2) = -1 - 4x$
   • Solve for $y$: $y = \frac{-1 - 4x}{3x - 2}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{-1 - 4x}{3x - 2}$

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

Here are five related questions to explore:

1. How do you verify that a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. How do you graph a function and its inverse on the same plane?
4. What does the inverse of a function represent geometrically?
5. How does the domain and range of a function change with its inverse?

Tip: Always remember that a function must be one-to-one (bijective) to have an inverse, meaning it passes both the vertical and horizontal line tests.