Question

Let f be the function defined by f(x)=(x^(3)-2x^(2)-3x)/(x^(3)-3x^(2)+4). Which of the following statements about f at x=2 and x=-1 is true?

Answer 1

At x=2, function f(x) displays a discontinuity as f(2) results in division by zero, indicating a vertical asymptote. For x=-1, the function yields an indeterminate form of 0/0, and further analysis would be necessary to understand its behavior at this point.

Step-by-step explanation:

The function f(x) is defined as f(x) = (x3 - 2x2 - 3x) / (x3 - 3x2 + 4). To understand the behavior of f at x = 2 and x = -1, we need to evaluate the function at these points, if possible. The continuity of the function and whether there are any discontinuities or indeterminate forms such as 0/0 at these points must be determined.

For x = 2, we substitute into the function:
f(2) = (23 - 2 × 22 - 3 × 2) / (23 - 3 × 22 + 4) = (8 - 8 - 6) / (8 - 12 + 4). This simplifies to f(2) = -6 / 0, which indicates a vertical asymptote or discontinuity at x = 2, as division by zero is undefined.

For x = -1, we substitute into the function:
f(-1) = ((-1)3 - 2 × (-1)2 - 3 × (-1)) / ((-1)3 - 3 × (-1)2 + 4) = (-1 - 2 + 3) / (-1 - 3 + 4), resulting in f(-1) = 0 / 0, which is an indeterminate form. More analysis, such as factoring or using L'Hôpital's Rule, would be needed to fully understand the behavior of f at x = -1.