The problem shown in the image asks to find the inverse of the function $f(x) = \frac{8}{2x - 3}$.

Steps to find the inverse function:

1. Start with the function $y = \frac{8}{2x - 3}$.

2. Swap $y$ and $x$ to find the inverse: $x = \frac{8}{2y - 3}$

3. Solve for $y$:

   • Multiply both sides by $2y - 3$ to get rid of the denominator: $x(2y - 3) = 8$

   • Distribute $x$ on the left-hand side: $2xy - 3x = 8$

   • Add $3x$ to both sides to isolate the $2xy$ term: $2xy = 8 + 3x$

   • Divide both sides by $2x$ to solve for $y$: $y = \frac{8 + 3x}{2x}$

   • This can be simplified to: $y = \frac{8}{2x} + \frac{3x}{2x} = \frac{4}{x} + \frac{3}{2}$

4. Therefore, the inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{4}{x} + \frac{3}{2}$

Would you like further details on this solution, or do you have any questions? Here are some related questions that can help expand your understanding:

1. What is the general process for finding the inverse of a function?
2. How do you verify if the inverse function found is correct?
3. How does the domain of the original function affect the inverse?
4. What happens to the graph of a function when you find its inverse?
5. What are some functions that do not have an inverse? Why?

Tip: Always remember that not all functions have inverses. A function must be one-to-one (pass the Horizontal Line Test) to have an inverse.