Question

Find the vertical asymptote(s) of f(x) = (x² - 3x - 6)/(x² - 4).

a) x = -1, 2
b) x = -2, 2
c) x = 1, -2
d) x = -1, 1

Answer 1

The function f(x) = (x² - 3x - 6)/(x² - 4) has vertical asymptotes at x = -2 and x = 2, because these values make the denominator zero without making the numerator zero.

Step-by-step explanation:

The student has asked to find the vertical asymptote(s) of the function f(x) = (x² - 3x - 6)/(x² - 4).

Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is not zero at those points. Therefore, we set the denominator equal to zero and solve:

x² - 4 = 0
±√(4) = x
x = 2 or x = -2

The function will not have a vertical asymptote at values which make the numerator zero at the same time. In this case, (x² - 3x - 6) does not equal zero when x = 2 or x = -2, so both values are indeed vertical asymptotes.

Therefore, the correct answer is b) x = -2, 2.