Question

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

\\f(x) = \frac{x^2 - 16}{x - 4}\\

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

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

Explanation:

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<reasoning>

Factor the rational function

Using the Rational Function Holes knowledge point
\[
f(x) = \frac{x^2 - 16}{x - 4} = \frac{(x - 4)(x + 4)}{x - 4}
\]

Determine domain exclusions

Using the Domain Exclusion knowledge point
\[
x - 4
eq 0 \implies x
eq 4
\]

Identify holes

Using the Rational Function Holes knowledge point
\[
\lim_{x \to 4} f(x) = \lim_{x \to 4} (x + 4) = 8
\]
Since the common factor \(x - 4\) cancels, a hole exists at \(x = 4\).

Determine vertical asymptotes

Using the Vertical Asymptotes Determination knowledge point
\[
\text{No remaining factors in denominator} \implies \text{No vertical asymptotes}
\]

Match the correct option

Using the Rational Function Holes and Vertical Asymptotes Determination knowledge points
\[
\text{Option A: No vertical asymptotes, hole at } x = 4
\]
</reasoning>

<answer>
<mcq-correct>A. There are no vertical asymptotes but there is (are) hole(s) corresponding to x = 4</mcq-correct>
<mcq-option>B. Vertical asymptote(s) at x = [blank] and hole(s) corresponding to x = [blank]</mcq-option>
<mcq-option>C. Vertical asymptote(s) at x = [blank]. There are no holes</mcq-option>
<mcq-option>D. There are no discontinuities</mcq-option>
</answer>

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Answer:

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<reasoning>

Factor the rational function

Using the Rational Function Holes knowledge point
\[
f(x) = \frac{x^2 - 16}{x - 4} = \frac{(x - 4)(x + 4)}{x - 4}
\]

Determine domain exclusions

Using the Domain Exclusion knowledge point
\[
x - 4
eq 0 \implies x
eq 4
\]

Identify holes

Using the Rational Function Holes knowledge point
\[
\lim_{x \to 4} f(x) = \lim_{x \to 4} (x + 4) = 8
\]
Since the common factor \(x - 4\) cancels, a hole exists at \(x = 4\).

Determine vertical asymptotes

Using the Vertical Asymptotes Determination knowledge point
\[
\text{No remaining factors in denominator} \implies \text{No vertical asymptotes}
\]

Match the correct option

Using the Rational Function Holes and Vertical Asymptotes Determination knowledge points
\[
\text{Option A: No vertical asymptotes, hole at } x = 4
\]
</reasoning>

<answer>
<mcq-correct>A. There are no vertical asymptotes but there is (are) hole(s) corresponding to x = 4</mcq-correct>
<mcq-option>B. Vertical asymptote(s) at x = [blank] and hole(s) corresponding to x = [blank]</mcq-option>
<mcq-option>C. Vertical asymptote(s) at x = [blank]. There are no holes</mcq-option>
<mcq-option>D. There are no discontinuities</mcq-option>
</answer>

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