which graph shows function $f$?

\f(x) = \begin{cases} (x + 5)^2 + 2, & x \le -4 \\ 2(3)^x - 1, & -4

Question

which graph shows function $f$?

\f(x) = \begin{cases} (x + 5)^2 + 2, & x \le -4 \\ 2(3)^x - 1, & -4 < x < 1 \\ -x + 8, & x \ge 1 \end{cases}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Piecewise Functions" ], "new_concepts": [], "current_concepts": [ "Piecewise Functions", "Function Comparison" ] } ## Evaluate the piecewise function at boundary points $$ \begin{aligned} &\text{For } x \le -4: \quad f(-4) = (-4 + 5)^2 + 2 = 1^2 + 2 = 3 \quad (\text{solid circle at } (-4, 3)) \ &\text{For } -4 < x < 1: \quad \lim_{x \to -4^+} f(x) = 2(3)^{-4} - 1 = \frac{2}{81} - 1 \approx -0.98 \quad (\text{open circle near } (-4, -1)) \ &\text{For } -4 < x < 1: \quad \lim_{x \to 1^-} f(x) = 2(3)^1 - 1 = 5 \quad (\text{open circle at } (1, 5)) \ &\text{For } x \ge 1: \quad f(1) = -1 + 8 = 7 \quad (\text{solid circle at } (1, 7)) \end{aligned} $$ ## Compare calculated points with the given graphs $$ \begin{aligned} &\text{Graph W: } \text{solid circle at } (-4, 3), \text{ open circle at } (-4, -1), \text{ open circle at } (1, 5), \text{ solid circle at } (1, -7) \quad (\text{Incorrect}) \ &\text{Graph X: } \text{solid circle at } (-4, -3), \text{ open circle at } (-4, -1) \quad (\text{Incorrect}) \ &\text{Graph Y: } \text{open circle at } (-4, 3), \text{ solid circle at } (-4, -1) \quad (\text{Incorrect}) \ &\text{Graph Z: } \text{solid circle at } (-4, 3), \text{ open circle at } (-4, -1), \text{ open circle at } (1, 5), \text{ solid circle at } (1, 7) \quad (\text{Correct}) \end{aligned} $$

(W) Graph W

(X) Graph X

(Y) Graph Y

(Z) Graph Z

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Piecewise Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the piecewise function at boundary points

Compare calculated points with the given graphs

Answer: