Question

For which of the following intervals is the function (in the picture) continuous? A. [−2.5,−1.5] B. [−1.5,−0.5] C. [−0.5,0.5] D. [0.5,1.5] E. [1.5,2.5]

Answer 1

Continuity of a Function

Given the function:

$f(x)=\frac{x+2}{x^4-2x^3-x^2+2x}$

It's required to select one of the intervals for continuity.

To analyze the function's continuity, we need to factorize the denominator as follows:

$f(x)=\frac{x+2}{x(x-1)(x+1)(x-2)}$

The denominator cannot be zero, thus each root of the polynomial in the denominator is a point of discontinuity. Listing them from least to greatest:

x = -1, x = 0, x = 1, x = 2

Any interval that contains one or more of the above-listed points is not an interval of continuity. Let's analyze them:

A. [-2.5, -1.5]

Both endpoints lie to the left of x = -1, thus the function is continuous in the interval

B. [-1.5, -0.5]

This interval contains the point x = -1, thus the function is not continuous here.

C. [-0.5, 0.5]

This interval contains the point x = 0, thus the function is not continuous here.

D. [1.5, 2.5]

This interval contains the point x = 2, thus the function is not continuous here.

Answer: A. [-2.5, -1.5]