1. **State the problem:** Simplify and analyze the expression $$\sqrt{\frac{X+3}{2}} - X$$. 2. **Recall the formula and rules:** The square root function $$\sqrt{y}$$ is defined for $$y \geq 0$$. Here, the expression inside the root is $$\frac{X+3}{2}$$, so we require $$\frac{X+3}{2} \geq 0$$ which implies $$X \geq -3$$. 3. **Rewrite the expression:** $$ \sqrt{\frac{X+3}{2}} - X = \frac{\sqrt{X+3}}{\sqrt{2}} - X $$ 4. **Domain:** Since $$X \geq -3$$, the expression is defined for all $$X$$ in $$[-3, \infty)$$. 5. **Further analysis:** To understand behavior, consider evaluating or plotting for values in the domain. For example, at $$X = -3$$: $$ \sqrt{\frac{-3+3}{2}} - (-3) = \sqrt{0} + 3 = 3 $$ At $$X=0$$: $$ \sqrt{\frac{0+3}{2}} - 0 = \sqrt{\frac{3}{2}} \approx 1.2247 $$ At $$X=2$$: $$ \sqrt{\frac{2+3}{2}} - 2 = \sqrt{\frac{5}{2}} - 2 \approx 1.5811 - 2 = -0.4189 $$ 6. **Summary:** The expression decreases as $$X$$ increases, starting from 3 at $$X=-3$$ and becoming negative for larger $$X$$. **Final answer:** The simplified form is $$\frac{\sqrt{X+3}}{\sqrt{2}} - X$$ with domain $$X \geq -3$$.