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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.

Learn how to find the inverse of the function f(x) = 8/(2x - 3) with a detailed step-by-step solution. This problem covers key algebraic concepts and inverse functions, suitable for students in Grades 10-12.

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