Question

Which statement is true about the discontinuities of the function f(x)? f (x) = StartFraction x squared minus 4 Over x cubed minus x squared minus 2 x EndFraction There is a hole at x = 2. There are asymptotes at x = 0 and x = –1. There are asymptotes at x = 0 and x = –1 and a hole at (2, two-thirds). There are holes at x = 0 and x = –1 and an asymptote at x = 2.

Answer 1

The function has asymptotes at x = 0 and x = -1, and a hole at (2, two-thirds).

Step-by-step explanation:

The statement that is true about the discontinuities of the function f(x) = \frac{{x^2 - 4}}{{x^3 - x^2 - 2x}} is that there are asymptotes at x = 0 and x = –1 and a hole at (2, two-thirds).

To determine this, we need to analyze the function's denominator. The denominator has factors of x - 2 and (x + 1). These factors indicate that there will be a hole at x = 2 and asymptotes at x = 0 and x = –1. The hole occurs at (2, two-thirds) because when we plug in x = 2 into the function, the undefined value at the denominator reduces to 2 - 2 = 0, giving us a hole at that point.