Question

Which statement about the asymptotes is true with respect to the graph of this function?

Answer 1

When you graph the equation you notice that there is 2 vertical and 1 horizontal asymptote. or D

Answer 2

Answer:

Option D.

Step-by-step explanation:

The given function is

$f\left(x\right)=\dfrac{3x^{2}-3}{x^{2}-4}$

In this function the degree of numerator an denominator is same i.e., 2.

Horizontal Asymptotes: If the degree of numerator an denominator is same, then

$\text{Horizontal asymptote}=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}$

$\text{Horizontal asymptote}=\frac{3}{1}$

$\text{Horizontal asymptote}=3$

Horizontal asymptote is y=3.

$f(x)=3$

$\dfrac{3x^{2}-3}{x^{2}-4}=3$

$3(x^{2}-1)=3(x^{2}-4)$

$x^{2}-1=x^{2}-4$

$1=4$

This statement is false for any value of x, therefore the graph does not cross the horizontal asymtote.

Vertical Asymptotes: Equate the denominator equal to 0, to find the vertical asymptotes.

$x^2-4=0$

Add 4 on both sides.

$x^2=4$

Taking square root on both sides.

$x=\pm \sqrt{4}$

$x=\pm 2$

Vertical asymptotes are x=2 and x=-2.

The graph has two vertical asymptotes and one horizontal asymptote.

Therefore, the correct option is D.