Question

the function $f(x) = \frac{1}{x^2 - 9}$ is a rational function. answer parts (a) - (i). (simplify your answer. type an equation. use integers or fractions for any numbers in the equation.) b. the function has two vertical asymptotes. the leftmost asymptote is $x = -3$ and the rightmost asymptote is $x = 3$. (simplify your answers. type equations. use integers or fractions for any numbers in the equation.) c. the function has no vertical asymptotes. g. find the equation(s) of all horizontal asymptotes or slant asymptotes. select the correct choice and, if necessary, fill in the answer box to complete your cho a. the horizontal asymptote is \boxed{}. (type an equation.) b. the function has no horizontal asymptote. it has a slant asymptote of \boxed{}. (type an equation. type your answer in slope-intercept form.) c. the function has no horizontal asymptote and has no slant asymptote.

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator and \( D(x) \) is the denominator:

• If the degree of \( N(x) \) (\( n \)) is less than the degree of \( D(x) \) (\( m \)), the horizontal asymptote is \( y = 0 \).
• If \( n = m \), the horizontal asymptote is \( y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)} \).
• If \( n>m \), there is no horizontal asymptote (may have a slant asymptote if \( n = m + 1 \)).

Step2: Analyze the Given Function

The function is \( f(x)=\frac{1}{x^{2}-9} \). Here, the numerator \( N(x)=1 \) (degree \( n = 0 \)) and the denominator \( D(x)=x^{2}-9 \) (degree \( m = 2 \)). Since \( n=0

Answer:

A. The horizontal asymptote is \( y = 0 \)