1. **State the problem:** Find the inverse of the equation $$5y + 4 = (x + 3)^2 + \frac{1}{2}$$. 2. **Rewrite the equation:** $$5y + 4 = (x + 3)^2 + \frac{1}{2}$$ 3. **Isolate the squared term:** $$ (x + 3)^2 = 5y + 4 - \frac{1}{2} = 5y + \frac{7}{2} $$ 4. **Express $x$ in terms of $y$ by taking the square root:** $$ x + 3 = \pm \sqrt{5y + \frac{7}{2}} $$ 5. **Solve for $x$:** $$ x = -3 \pm \sqrt{5y + \frac{7}{2}} $$ 6. **Swap $x$ and $y$ to find the inverse function:** $$ y = -3 \pm \sqrt{5x + \frac{7}{2}} $$ 7. **Rewrite the constant inside the square root as a fraction:** $$ y = -3 \pm \sqrt{5x + \frac{7}{2}} $$ 8. **Compare with the given options:** The option matching this form is: $$ y = -3 \pm \sqrt{5x + \frac{7}{2}} $$ **Final answer:** $$ y = -3 \pm \sqrt{5x + \frac{7}{2}} $$