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 the choice.
a. the x-intercept(s) is \\( \boxed{0} \\).
(use 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 the choice.
a. the equation(s) of the vertical asymptote(s) is/are \\( \boxed{} \\).
(use 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, we set \( f(x)=0 \). Given \( f(x)=\frac{3x}{x^{2}-4} \), we set \( \frac{3x}{x^{2}-4}=0 \). A fraction is zero when the numerator is zero (and the denominator is not zero). So, solve \( 3x = 0 \), which gives \( x = 0 \). We check the denominator at \( x = 0 \): \( 0^{2}-4=- 4
eq0 \), so \( x = 0 \) is a valid x - intercept.

Step2: Find vertical asymptotes

To find vertical asymptotes, we set the denominator equal to zero and solve for \( x \). The denominator is \( x^{2}-4 \), which factors as \( (x - 2)(x + 2) \). Setting \( (x - 2)(x + 2)=0 \), we get \( x-2 = 0 \) or \( x + 2=0 \), so \( x = 2 \) or \( x=-2 \). These are the equations of the vertical asymptotes.

Answer:

• x - intercept: \( 0 \)
• Vertical asymptotes: \( x = 2,x=-2 \)