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{ "quality": "incomplete", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Domain of a Function", "Increasing and Decreasing Functions" ], "new_concepts": [ "Piecewise Functions", "Function Comparison" ], "current_concepts": [ "Domain of a Function", "Increasing and Decreasing Functions", "Piecewise Functions", "Function Comparison" ] } ## Analyze the graph of function $f$ $$ \begin{aligned} &\text{Domain of } f: (-\infty, \infty)\ &\text{Vertex/Peak of } f: (1, 3)\ &\text{For } x \le 1: f(x) \text{ is a line passing through } (-1, 0), (0, 2), (1, 3) \implies f(x) = x + 2\ &\text{For } x > 1: f(x) \text{ is a line passing through } (1, 3), (4, 0), (7, -3) \implies f(x) = -x + 4 \end{aligned} $$ ## Analyze the piecewise function $g(x)$ $$ g(x) = \begin{cases} x - 4, & x \le 1 \ -3, & 1 < x \le 3 \ x - 6, & x > 3 \end{cases} $$ ## Compare key features of $f(x)$ and $g(x)$ $$ \begin{aligned} &\text{y-intercept of } f: f(0) = 2\ &\text{y-intercept of } g: g(0) = 0 - 4 = -4\ &\text{x-intercepts of } f: x = -2 \text{ and } x = 4\ &\text{x-intercepts of } g:\ &\quad \text{For } x \le 1: x - 4 = 0 \implies x = 4 \quad (\text{not in domain } x \le 1)\ &\quad \text{For } x > 3: x - 6 = 0 \implies x = 6 \quad (\text{valid since } 6 > 3)\ &\quad \implies \text{x-intercept of } g \text{ is } x = 6 \end{aligned} $$ ## Question 1 (A) The y-intercept of $f(x)$ is greater than the y-intercept of $g(x)$.

(B) The y-intercept of $f(x)$ is less than the y-intercept of $g(x)$.

(C) The x-intercept of $g(x)$ is $x = -4$.

(D) Both functions have the same y-intercept.

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

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Explanation:

Analyze the graph of function \(f\)

Analyze the piecewise function \(g(x)\)

Compare key features of \(f(x)\) and \(g(x)\)

Answer:

Question 1