Question

For which intervals is the function positive? Select each correct answer. (-6,-2) (4,8] [-8,-6) (-2,4)

Answer 1

The function is positive in the intervals (-6, -2) and (4, 8].

Step-by-step explanation:

The given function is positive where its graph is above the x-axis. To determine this, we examine the function's sign over different intervals.

Starting with the interval (-6, -2), we substitute any value within this range into the function. If the result is positive, then the function is positive in that interval. Similarly, we evaluate the function for the interval (4, 8]. If the output is positive, the function is positive in this range.

For the interval (-6, -2), consider a test point, say -4. Substitute -4 into the function, and if the result is positive, then the function is positive in this interval. Repeat this process for the interval (4, 8] using a test point like 6.

In contrast, for the intervals [-8, -6) and (-2, 4), if the function's value is negative for any test point within these intervals, then the function is not positive in those ranges. Therefore, we exclude [-8, -6) and (-2, 4) from our final answer. This method of testing intervals helps identify where the function is positive and provides a clear rationale for the chosen intervals.

Answer 2

The function is positive on intervals (-6, -2) and (4, 8]. It remains above zero in these ranges, while other intervals may have non-positive values.

Step-by-step explanation:

The function's positivity is determined by examining where it stays above the x-axis. In the intervals (-6, -2) and (4, 8], the function consistently produces positive values. These intervals were chosen based on the behavior of the function, ensuring positivity throughout.

However, intervals like [-8, -6) and (-2, 4) don't guarantee positivity, as the function may cross the x-axis, leading to non-positive values within those ranges. The inclusion or exclusion of endpoints is also considered for accuracy in defining intervals.

This analysis provides a clear understanding of the intervals where the function exhibits positivity.