Question

list all the asymptotes of the following function, then graph the function.\\( f(x) = \frac{5x}{x^2 - 9} \\)\
choose the correct asymptotes below.\
\\( \bigcirc \\) a. \\( x = 5, x = -5, y = 0 \\)\
\\( \bigcirc \\) b. \\( x = 3, x = -3, y = 0 \\)\
\\( \bigcirc \\) c. \\( x = 5, x = -5 \\)\
\\( \bigcirc \\) d. \\( x = 3, x = -3 \\)

Explanation:

Step1: Find Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 - 9 \), so:
\( x^2 - 9 = 0 \)
Factor the left side (difference of squares):
\( (x - 3)(x + 3) = 0 \)
Set each factor equal to zero:
\( x - 3 = 0 \) or \( x + 3 = 0 \)
Solving these gives \( x = 3 \) or \( x = -3 \). These are the vertical asymptotes.

Step2: Find Horizontal Asymptote

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), if the degree of \( N(x) \) (degree of numerator) is less than the degree of \( D(x) \) (degree of denominator), the horizontal asymptote is \( y = 0 \). Here, the numerator \( 5x \) has degree 1, and the denominator \( x^2 - 9 \) has degree 2. Since \( 1 < 2 \), the horizontal asymptote is \( y = 0 \).

Answer:

B. \( x = 3, x = -3, y = 0 \)