Final answer:
To find intervals where f(x) = 9 - x² / (x² - 4) is positive, one must assess the signs of the numerator and denominator separately. The function is positive in intervals where either both the numerator and denominator are positive, or both negative, leading to the correct intervals being (-3, -2) and (2, 3).
Step-by-step explanation:
To determine for which intervals the function f(x) = 9 - x² / (x² - 4) is positive, we need to consider the sign of the numerator and the denominator separately. The numerator 9 - x² is positive when x² < 9, which is true for all x in the interval (-3, 3). The denominator x² - 4 is positive when x² > 4, which is true for |x| > 2. Hence, the function is positive when both the numerator and denominator are positive, and also when they are both negative.
First, the function is positive when x is in (-3, -2) because both the numerator and denominator are negative, resulting in a positive value for f(x). Next, we look for intervals where both are positive: this happens for x in (2, 3). Therefore, the correct intervals where f(x) is positive are options b) (-3, -2) and d) (2, 3). Option e) (0, 3) cannot be correct as it contains the interval (0, 2) where the denominator is negative, while the numerator is positive.