Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\frac{8 x}{x^{2}-1} \) Find any vertical asymptotes of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is \( \square \) and the rightmost asymptote is (Type equations.) C. There are no vertical asymptotes. Find the intervals where \( f(x) \) is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is increasing on \( \square \). It is never decreasing. B. The function is increasing on \( \square \). It is decreasing on \( \square \). C. The function is decreasing on \( \square \). It is never increasing.

To analyze the function \( f(x) = \frac{8x}{x^2 - 1} \), we start by identifying the vertical asymptotes. The denominator \( x^2 - 1 \) can be factored to \( (x - 1)(x + 1) \). Thus, the function will have vertical asymptotes where the denominator equals zero, which occurs at \( x = 1 \) and \( x = -1 \). So, the correct answer is B: The function has two vertical asymptotes. The leftmost asymptote is \( x = -1 \) and the rightmost asymptote is \( x = 1 \). Next, we assess where the function is increasing or decreasing. By finding the derivative \( f'(x) \) and analyzing its sign, we can determine the intervals. The function \( f(x) \) will be increasing in the intervals where \( f'(x) > 0 \) and decreasing where \( f'(x) < 0 \). After performing this analysis, we find the function increasing on \( (-\infty, -1) \cup (1, \infty) \) and decreasing on \( (-1, 1) \). Thus, the correct choice is B: The function is increasing on \( (-\infty, -1) \cup (1, \infty) \). It is decreasing on \( (-1, 1) \).