Question: Lesson 19 Practice Set 1
15. Find the domain and range for the function shown, and state whether it is continuous. If continuous, identify the x-location of any local or global extrema.
Select one:
Discontinuous, Domain: $-5 \leq x \leq 6$; Range: $-2 \leq f(x) \leq 4$
Discontinuous, Domain: $-5 \leq x < 6$; Range: $-2 \leq f(x) \leq 4$ Global max at $x = 1$, min at $x = -5$
Continuous, Domain: $-5 \leq x < 6$; Range: $-2 \leq f(x) \leq 4$ Global max at $x = -1$, min at $x = -5$; Local max or
Continuous, Domain: $-5 \leq x \leq 6$; Range: $-2 \leq f(x) < 4$ Global max at $x = -1$, min at $x = -5$; Local max or
Graph: A smooth curve labeled $f(x)$ appears in the top-right of the question area; it starts at a filled point near $x = -5$, $y = -2$, rises to a local/global peak around $x = -1$, $y = 4$, then falls to a local minimum around $x = 4$, $y = 0$, and rises again to a filled point near $x = 6$, $y = 1$. position_hint=top-right
1. **State the problem:** We need to find the domain and range of the function $f(x)$ shown in the graph, determine if it is continuous, and identify any local or global extrema.
2. **Analyze the domain:** The graph starts at $x = -5$ with a filled point, meaning the function is defined at $x = -5$. It ends at $x = 6$ with a filled point, so the function is defined at $x = 6$ as well. The curve is smooth with no breaks, so the domain is $-5 \leq x \leq 6$.
3. **Analyze the range:** The lowest point on the graph is at $y = -2$ (at $x = -5$), and the highest point is at $y = 4$ (at $x = -1$). The function reaches $y = 4$ exactly, so the range includes $4$. The function also reaches $y = -2$ exactly, so the range includes $-2$. The range is therefore $-2 \leq f(x) \leq 4$.
4. **Check continuity:** The graph is a smooth curve with no breaks or holes, so the function is continuous on the domain $-5 \leq x \leq 6$.
5. **Identify extrema:**
- Global maximum is at $x = -1$ where $f(x) = 4$.
- Global minimum is at $x = -5$ where $f(x) = -2$.
- There is a local minimum around $x = 4$ where $f(x) = 0$.
6. **Conclusion:** The function is continuous with domain $-5 \leq x \leq 6$, range $-2 \leq f(x) \leq 4$, global max at $x = -1$, global min at $x = -5$, and a local minimum near $x = 4$.
**Final answer:** Continuous, Domain: $-5 \leq x \leq 6$; Range: $-2 \leq f(x) \leq 4$; Global max at $x = -1$, min at $x = -5$; Local min near $x = 4$.