\\begin{cases} 1 - 2x, & \text{if } x \le 2 \\ 5x - 4, & \text{if } x 2 \end{cases}\

Question

\\begin{cases} 1 - 2x, & \text{if } x \le 2 \\ 5x - 4, & \text{if } x > 2 \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", "Evaluating Piecewise Functions" ], "new_concepts": [], "current_concepts": [ "Piecewise Functions", "Evaluating Piecewise Functions", "Piecewise Linear Functions" ] } ## Evaluate the first piece at the boundary $$ \begin{aligned} &f(x) = 1 - 2x \quad \text{for } x \le 2\ &f(2) = 1 - 2(2) = -3\ &\text{Since the inequality is } \le \text{, the point } (2, -3) \text{ must be a solid (closed) circle.} \end{aligned} $$ ## Evaluate the second piece at the boundary $$ \begin{aligned} &f(x) = 5x - 4 \quad \text{for } x > 2\ &\lim_{x \to 2^+} f(x) = 5(2) - 4 = 6\ &\text{Since the inequality is } > \text{, the point } (2, 6) \text{ must be an open circle.} \end{aligned} $$ ## Identify the matching graph $$ \begin{aligned} &\text{First graph: open circle at } (2, -3) \text{ and solid circle at } (2, 6) \quad (\text{Incorrect})\ &\text{Second graph: solid circle at } (2, -3) \text{ and open circle at } (2, 6) \quad (\text{Correct}) \end{aligned} $$

(A) Graph with an open circle at $(2, -3)$ and a solid circle at $(2, 6)$

(B) Graph with a solid circle at $(2, -3)$ and an open circle at $(2, 6)$

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

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QUESTION IMAGE

Question

Explanation:

Evaluate the first piece at the boundary

Evaluate the second piece at the boundary

Identify the matching graph

Answer: