Question

State the intervals in which the reciprocal function of y =(x + 3)²⁻¹ is increasing or decreasing. Sketch the reciprocal function.

Answer 1

The reciprocal function of y =(x + 3)²⁻¹ is increasing on the intervals (-∞, -3) U (-3, ∞) and there are no decreasing intervals. To sketch the reciprocal function, plot points based on these intervals and connect them with a smooth curve.

Step-by-step explanation:

The reciprocal function of y =(x + 3)²⁻¹ can be found by taking the reciprocal of the original function. So, the reciprocal function is y' = 1/(x + 3)². To determine the intervals where the reciprocal function is increasing or decreasing, we need to find the intervals where the derivative of the reciprocal function is positive or negative.

1. Increasing Intervals:To find the increasing intervals, we need to find where the first derivative of the reciprocal function is positive. Taking the derivative, we get y' = -2/(x + 3)³. Setting this derivative to be greater than zero, we have -2/(x + 3)³ > 0. Solving this inequality, we find that the reciprocal function is increasing on the interval (-∞, -3) U (-3, ∞).
2. Decreasing Intervals:To find the decreasing intervals, we need to find where the first derivative of the reciprocal function is negative. Taking the derivative, we get y' = -2/(x + 3)³. Setting this derivative to be less than zero, we have -2/(x + 3)³ < 0. Solving this inequality, we find that the reciprocal function is decreasing on the interval empty set (i.e., there are no decreasing intervals).

To sketch the reciprocal function, we can plot some points using the given intervals and then connect them with a smooth curve. In this case, since there are no decreasing intervals, the reciprocal function will only have one branch.