1. We are asked to evaluate the expression $2 + \sqrt{x+4}$ for values of $x$ between $-3$ and $12$ inclusive. 2. First, check the expression inside the square root: $$x+4$$ It must be non-negative for the square root to be defined in real numbers. 3. Since $x \geq -3$, then $$x+4 \geq -3 + 4 = 1$$ so the expression inside the root is always positive for $x$ in this range. 4. To understand the values of the expression over the interval, evaluate at the endpoints: - At $x = -3$: $$2 + \sqrt{-3 + 4} = 2 + \sqrt{1} = 2 + 1 = 3$$ - At $x = 12$: $$2 + \sqrt{12 + 4} = 2 + \sqrt{16} = 2 + 4 = 6$$ 5. Since the square root function is increasing, the entire expression $2 + \sqrt{x+4}$ is increasing on the interval $[-3, 12]$. **Final answer:** The expression $2 + \sqrt{x+4}$ ranges from 3 to 6 as $x$ goes from $-3$ to $12$.