1. **Problem:** Find the inverse function $f^{-1}(x)$ of $f(x) = \frac{3x - 7}{4x + 3}$. 2. **Formula and rules:** To find the inverse function, swap $x$ and $y$ in the equation $y = f(x)$ and solve for $y$. 3. **Step-by-step solution:** - Start with $y = \frac{3x - 7}{4x + 3}$. - Swap $x$ and $y$: $x = \frac{3y - 7}{4y + 3}$. - Multiply both sides by $4y + 3$: $x(4y + 3) = 3y - 7$. - Distribute $x$: $4xy + 3x = 3y - 7$. - Group terms with $y$ on one side: $4xy - 3y = -3x - 7$. - Factor out $y$: $y(4x - 3) = -3x - 7$. - Solve for $y$: $$y = \frac{-3x - 7}{4x - 3}$$. 4. **Answer:** The inverse function is $$f^{-1}(x) = \frac{-3x - 7}{4x - 3}$$. This method applies similarly to other rational functions: swap variables and solve for the new dependent variable.