Question

follow the seven step strategy to graph the following rational function.
$f(x) = -\frac{2}{x^2 - 9}$

what is/are the x-intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercept(s) is/are \\(\boxed{}\\).
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. there is no x-intercept.

find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the vertical asymptote(s) is/are \\(\boxed{}\\).
(type an equation. use a comma to separate answers as needed.)
b. there is no vertical asymptote.

Explanation:

Step1: Find x-intercept

To find the x-intercept, set \( f(x) = 0 \). So we have the equation \( -\frac{2}{x^2 - 9}=0 \). Multiply both sides by \( x^2 - 9 \) (assuming \( x^2 - 9
eq0 \)), we get \( - 2=0 \), which is a contradiction. So there is no x-intercept.

Step2: Find vertical asymptote

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 - 9 \). Factor it as \( (x - 3)(x + 3) \). Set \( (x - 3)(x + 3)=0 \). Solving \( x - 3 = 0 \) gives \( x = 3 \), and solving \( x+3 = 0 \) gives \( x=-3 \). So the vertical asymptotes are \( x = 3 \) and \( x=-3 \).

For x - intercept:

To find the x - intercept, we set \( f(x)=0 \). The function is \( f(x)=-\frac{2}{x^{2}-9} \). Setting \( -\frac{2}{x^{2}-9} = 0 \) leads to \( - 2=0 \) (after multiplying both sides by \( x^{2}-9 \), \( x^{2}
eq9 \)), which is impossible. So there is no x - intercept.

Vertical asymptotes of a rational function occur where the denominator is zero (and the numerator is not zero at those points). The denominator of \( f(x)=-\frac{2}{x^{2}-9} \) is \( x^{2}-9=(x - 3)(x + 3) \). Setting \( (x - 3)(x + 3)=0 \) gives \( x = 3 \) or \( x=-3 \). The numerator \( - 2
eq0 \) at these values, so these are vertical asymptotes.

Answer:

B. There is no x - intercept.

For vertical asymptote: