graph the function $f(x) = \frac{1}{x^2 - x}$ using a graphing utility with the window $-1, 2 \times -10, 10$. use the graph to determine the following limits. complete parts a through d below.

first, graph the function using a graphing utility. choose the correct graph below.

Question

graph the function $f(x) = \frac{1}{x^2 - x}$ using a graphing utility with the window $-1, 2 \times -10, 10$. use the graph to determine the following limits. complete parts a through d below.

first, graph the function using a graphing utility. choose the correct graph below.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Vertical Asymptote Definition", "Infinite Limits" ], "new_concepts": [ "Rational Function Graphing" ], "current_concepts": [ "Vertical Asymptote Definition", "Infinite Limits", "Rational Function Graphing" ] } ## Analyze vertical asymptotes and domain $$ \begin{aligned} &f(x) = \frac{1}{x^2 - x} = \frac{1}{x(x-1)}\ &\text{Vertical asymptotes at } x = 0 \text{ and } x = 1 \end{aligned} $$ ## Determine behavior in intervals $$ \begin{aligned} &\text{For } x \in (0, 1): x(x-1) < 0 \implies f(x) < 0 \implies \lim_{x \to 0^+} f(x) = -\infty, \lim_{x \to 1^-} f(x) = -\infty\ &\text{For } x < 0: x(x-1) > 0 \implies f(x) > 0 \implies \lim_{x \to 0^-} f(x) = +\infty\ &\text{For } x > 1: x(x-1) > 0 \implies f(x) > 0 \implies \lim_{x \to 1^+} f(x) = +\infty \end{aligned} $$ ## Match behavior to graph options $$ \begin{aligned} &\text{Graph must have a downward-opening U-shape between } x = 0 \text{ and } x = 1\ &\text{Graph must go to } +\infty \text{ for } x < 0 \text{ and } x > 1\ &\text{This matches Option B} \end{aligned} $$

(A) U-shape opening upward between the asymptotes

(B) U-shape opening downward between the asymptotes, with curves going to positive infinity on the outer sides

(C) U-shape opening upward between the asymptotes, with curves going to negative infinity on the outer sides

(D) U-shape opening downward between the asymptotes, with curves going to negative infinity on the outer sides

{ "elements": [ { "type": "line", "params": [[0, -10], [0, 10]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2, "name": "x=0", "withLabel": false } }, { "type": "line", "params": [[1, -10], [1, 10]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2, "name": "x=1", "withLabel": false } }, { "type": "functiongraph", "params": [{"js": "1/(x*x - x)", "latex": "\frac{1}{x^2-x}"}, -1, -0.01], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [{"js": "1/(x*x - x)", "latex": "\frac{1}{x^2-x}"}, 0.01, 0.99], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "functiongraph", "params": [{"js": "1/(x*x - x)", "latex": "\frac{1}{x^2-x}"}, 1.01, 2], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } } ], "timestamps": [0.1] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Rational Function Graphing" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze vertical asymptotes and domain

Determine behavior in intervals

Match behavior to graph options

Answer: