Question

find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational function.

(f(x) = \frac{x}{x^2 + 17})

select the correct choice below and, if necessary, fill in the answer box to complete your choice. (type an integer or a fraction. use a comma to separate answers as needed.)

a. the vertical asymptote(s) is (are) (x =) . there are no holes.
b. there are no vertical asymptotes but there is (are) hole(s) corresponding to (x =) .
c. the vertical asymptote(s) is (are) (x =) and hole(s) corresponding to (x =) .
d. there are no discontinuities.

Explanation:

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<reasoning>

Analyze the denominator for domain exclusions

Using the Vertical Asymptotes Determination and Rational Function Holes knowledge points
\[

$$\begin{aligned} &x^2 + 17 = 0 \\ &x^2 = -17 \\ &x = \pm i\sqrt{17} \end{aligned}$$

\]

Determine real discontinuities

Using the Domain Exclusion knowledge point
\[

$$\begin{aligned} &\text{Since } x^2 + 17 \ge 17 \text{ for all real } x, \\ &x^2 + 17 e 0 \text{ for all } x \in \mathbb{R}. \end{aligned}$$

\]

Identify asymptotes and holes

Using the Vertical Asymptotes Determination and Rational Function Holes knowledge points
\[

$$\begin{aligned} &\text{No real values of } x \text{ make the denominator zero.} \\ &\text{Therefore, there are no vertical asymptotes and no holes.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. The vertical asymptote(s) is (are) \(x =\) [blank]. There are no holes.</mcq-option>
<mcq-option>B. There are no vertical asymptotes but there is (are) hole(s) corresponding to \(x =\) [blank].</mcq-option>
<mcq-option>C. The vertical asymptote(s) is (are) \(x =\) [blank] and hole(s) corresponding to \(x =\) [blank].</mcq-option>
<mcq-correct>D. There are no discontinuities.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Analyze the denominator for domain exclusions

Using the Vertical Asymptotes Determination and Rational Function Holes knowledge points
\[

$$\begin{aligned} &x^2 + 17 = 0 \\ &x^2 = -17 \\ &x = \pm i\sqrt{17} \end{aligned}$$

\]

Determine real discontinuities

Using the Domain Exclusion knowledge point
\[

$$\begin{aligned} &\text{Since } x^2 + 17 \ge 17 \text{ for all real } x, \\ &x^2 + 17 e 0 \text{ for all } x \in \mathbb{R}. \end{aligned}$$

\]

Identify asymptotes and holes

Using the Vertical Asymptotes Determination and Rational Function Holes knowledge points
\[

$$\begin{aligned} &\text{No real values of } x \text{ make the denominator zero.} \\ &\text{Therefore, there are no vertical asymptotes and no holes.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. The vertical asymptote(s) is (are) \(x =\) [blank]. There are no holes.</mcq-option>
<mcq-option>B. There are no vertical asymptotes but there is (are) hole(s) corresponding to \(x =\) [blank].</mcq-option>
<mcq-option>C. The vertical asymptote(s) is (are) \(x =\) [blank] and hole(s) corresponding to \(x =\) [blank].</mcq-option>
<mcq-correct>D. There are no discontinuities.</mcq-correct>
</answer>

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