find the vertical asymptotes, if any, of the graph of the rational function.

(f(x) = \frac{x - 4}{x(x - 2)})

a. (x = 2)
b. (x = 4) and (x = 2)
c. (x = 0) and (x = 2)
d. no vertical asymptote

Question

find the vertical asymptotes, if any, of the graph of the rational function.

(f(x) = \frac{x - 4}{x(x - 2)})

a. (x = 2)
b. (x = 4) and (x = 2)
c. (x = 0) and (x = 2)
d. no vertical asymptote

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Vertical Asymptotes Determination" ], "new_concepts": [], "current_concepts": [ "Rational Functions", "Vertical Asymptotes Determination" ] } ## Identify the rational function Using the *Vertical Asymptotes Determination* knowledge point $$ f(x) = \frac{x - 4}{x(x - 2)} $$ ## Find the domain restrictions Using the *Vertical Asymptotes Determination* knowledge point $$ \begin{aligned} &x(x - 2) = 0 \ &x = 0 \quad \text{or} \quad x = 2 \end{aligned} $$ ## Determine vertical asymptotes Using the *Vertical Asymptotes Determination* knowledge point $$ \begin{aligned} &\lim_{x \to 0} \frac{x - 4}{x(x - 2)} = \pm\infty \ &\lim_{x \to 2} \frac{x - 4}{x(x - 2)} = \pm\infty \end{aligned} $$ The vertical asymptotes are $x = 0$ and $x = 2$.

A. $x = 2$

B. $x = 4$ and $x = 2$

C. $x = 0$ and $x = 2$

D. no vertical asymptote

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Vertical Asymptotes Determination" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the rational function

Find the domain restrictions

Determine vertical asymptotes

Answer: