Question

find the equation of all horizontal asymptotes (if any) of the rational function.
$f(x)=\frac{9x^2}{8 - x^2}$
select the correct choice below and fill in any answer boxes within your choice.
a. the function has two horizontal asymptotes. the top asymptote is \\(\square\\), and the bottom asymptote is \\(\square\\). (simplify your answers. type equations. use integers or fractions for any numbers in the equations.)
b. the function has one horizontal asymptote. \\(\square\\) (simplify your answer. type an equation. use integers or fractions for any numbers in the equation.)
c. the function has no horizontal asymptotes.

Explanation:

Step1: Analyze degrees of numerator and denominator

The function is \( f(x)=\frac{9x^2}{8 - x^2} \). The degree of the numerator (highest power of \( x \)) is \( 2 \), and the degree of the denominator (after rearranging \( -x^2 + 8 \)) is also \( 2 \).

Step2: Find horizontal asymptote for equal degrees

When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is \( 9 \), and the leading coefficient of the denominator (for \( -x^2 + 8 \)) is \( -1 \). So the horizontal asymptote is \( y=\frac{9}{-1}=-9 \). Since there's only one horizontal asymptote (as the degrees are equal and we have one ratio), option B applies.

Answer:

B. The function has one horizontal asymptote. \( y = -9 \)