Question

follow the seven step strategy to graph the following rational function.
$f(x) = \frac{3x}{x^2 - 4}$

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 0.
(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 \square.
(type an equation. use a comma to separate answers as needed.)
b. there is no vertical asymptote.

Explanation:

Step1: Find Vertical Asymptotes

To find vertical asymptotes of a rational function \( f(x)=\frac{N(x)}{D(x)} \), we set the denominator \( D(x) = 0 \) (after ensuring numerator and denominator have no common factors). For \( f(x)=\frac{3x}{x^2 - 4} \), factor the denominator: \( x^2 - 4=(x - 2)(x + 2) \). Set \( (x - 2)(x + 2)=0 \).

Step2: Solve for x

Using the zero - product property, if \( (x - 2)(x + 2)=0 \), then \( x - 2 = 0 \) or \( x+2 = 0 \). Solving \( x - 2 = 0 \) gives \( x = 2 \), and solving \( x + 2 = 0 \) gives \( x=-2 \). There are no common factors between the numerator \( 3x \) and the denominator \( (x - 2)(x + 2) \), so these are the vertical asymptotes.

Answer:

The equation(s) of the vertical asymptote(s) is/are \( x = - 2,x = 2 \)