Final answer:
To establish where the function f(x) = x/(x² - 9) is increasing or decreasing, we need to calculate the first derivative and analyze its sign over the intervals (-∞, -3), (-3, 3), and (3, ∞). Vertical asymptotes occur at x = -3 and x = 3, creating intervals for sign chart analysis. The function is likely decreasing in the first interval, increasing in the second, and decreasing in the third.
Step-by-step explanation:
The student's question involves analyzing the behavior of the function f(x) = \frac{x}{x^2 - 9}. To determine whether the function is increasing or decreasing on the given intervals, we must look at the first derivative of the function. The intervals are (-∞, -3), (-3, 3), and (3, ∞). Since the function is not defined at x = -3 and x = 3 (as these values make the denominator zero), these points are vertical asymptotes.
Finding the first derivative will help us identify the intervals where the function's slope is positive (increasing) or negative (decreasing). By applying the quotient rule, we get f'(x). We can then use a sign chart to determine the sign of the derivative in each interval. If the derivative is positive on an interval, the function is increasing on that interval. If it is negative, the function is decreasing.
Upon examining the first derivative with a sign chart, one can determine the increasing or decreasing behavior of the given function across the specified intervals. For this particular function, we expect it to be decreasing in the first interval, increasing in the second, and decreasing again in the third interval. This is due to the negative sign in the denominator, which will change the sign of the slope as x crosses the asymptotes at -3 and 3.