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

Explanation:

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<reasoning>

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 \(x = 2\).
</reasoning>

<answer>
<mcq-option>A. \(x = 2\)</mcq-option>
<mcq-option>B. \(x = 4\) and \(x = 2\)</mcq-option>
<mcq-correct>C. \(x = 0\) and \(x = 2\)</mcq-correct>
<mcq-option>D. no vertical asymptote</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Vertical Asymptotes Determination"
]
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Answer:

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<reasoning>

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 \(x = 2\).
</reasoning>

<answer>
<mcq-option>A. \(x = 2\)</mcq-option>
<mcq-option>B. \(x = 4\) and \(x = 2\)</mcq-option>
<mcq-correct>C. \(x = 0\) and \(x = 2\)</mcq-correct>
<mcq-option>D. no vertical asymptote</mcq-option>
</answer>

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