Question: 7. y = \sqrt{-x - 2} + 8
Domain: ____________________
Range: ____________________
Graph shape: square-root curve reflected across the y-axis and shifted left 2 and up 8, starting at (-2, 8) and extending to the left and upward. position_hint \in \{top-right, center\}
1. **State the problem:** Find the domain and range of the function $$y = \sqrt{-x - 2} + 8$$ and describe its graph.
2. **Recall the domain rule for square root functions:** The expression inside the square root must be \textbf{non-negative} because the square root of a negative number is not real.
3. **Set the radicand \( -x - 2 \) \geq 0:**
$$
-x - 2 \geq 0
$$
4. **Solve for \(x\):**
$$
-x \geq 2
$$
$$
x \leq -2
$$
5. **Domain:**
$$
\boxed{(-\infty, -2]}
$$
6. **Find the range:**
- The square root function outputs values \( \geq 0 \).
- The smallest value inside the root is 0 (when \(x = -2\)), so the smallest output of the root is 0.
- Adding 8 shifts the entire function up by 8.
7. **Minimum value of \(y\):**
$$
y_{min} = 0 + 8 = 8
$$
8. **Range:**
$$
\boxed{[8, \infty)}
$$
9. **Graph shape explanation:**
- The function is a square root curve reflected across the y-axis because of the negative sign inside.
- It is shifted left by 2 units and up by 8 units.
- The graph starts at the point $$(-2, 8)$$ and extends to the left and upward.
**Final answers:**
- Domain: $$(-\infty, -2]$$
- Range: $$[8, \infty)$$