Question

Find the equations of the horizontal asymptotes and the vertical asymptotes of . If there are no asymptotes of a given type, enter NONE. If there is more than one asymptote of a given type, give a comma seperated list (i.e.: 1, 2,...).

f(x) = ( x^2 - 9) / (3x^2 + 6x - 9)

Horizontal asymptotes: y = ?
Vertical Asymptotes: x = ?

Answer 1

The horizontal asymptote of the given function is y = 1/3.

The vertical asymptotes are x = 1 and x = -3.

Step-by-step explanation:

To find the horizontal asymptotes of a function, we need to compare the degrees of the numerator and denominator polynomials.

In this case, the degree of the numerator is 2 and the degree of the denominator is 2 as well.

Since the degrees are equal, we can find the horizontal asymptote by dividing the leading coefficients of the numerator and denominator.

So, the horizontal asymptote of the given function is y = 1/3.

To find the vertical asymptotes, we need to solve the equation 3x^2 + 6x - 9 = 0.

=> 3 (x^2 + 2x - 3) = 0

=> ( x+3) ( x-1) = 0

Therefore, the equation has two solutions x = 1 and x = -3.

Therefore, the vertical asymptotes of the given function are x = 1 and x = -3.