1. **Problem:** Find the inverse of the function $f(x) = \frac{2x + 3}{2x - 3}$ where $x \neq \frac{3}{2}$. 2. **Formula and rules:** To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the equation $y = f(x)$ and solve for $y$. 3. **Step-by-step solution:** - Start with $y = \frac{2x + 3}{2x - 3}$. - Swap $x$ and $y$: $x = \frac{2y + 3}{2y - 3}$. - Multiply both sides by $(2y - 3)$: $x(2y - 3) = 2y + 3$. - Distribute $x$: $2xy - 3x = 2y + 3$. - Group terms with $y$ on one side: $2xy - 2y = 3x + 3$. - Factor out $y$: $y(2x - 2) = 3x + 3$. - Solve for $y$: $$y = \frac{3x + 3}{2x - 2} = \frac{3(x + 1)}{2(x - 1)}.$$ 4. **Answer:** The inverse function is $$f^{-1}(x) = \frac{3}{2} \cdot \frac{x + 1}{x - 1}.$$ 5. **Matching with options:** This corresponds to option C. --- **Final answer:** C. $\frac{3}{2} \cdot \frac{x + 1}{x - 1}$