Question

Consider the function f(x)= 9-x^2/ x^2 -4 .

For which intervals is f(x) positive? Check all that apply.

(–∞, –3)
(–3, –2)
(–2, 2)
(2, 3)
(∞, 3)

Answer 1

Interpret f(x) = (9 − x^2)/(x^2 − 4) = (3 − x)(3 + x)/((x − 2)(x + 2)). Zeros at x = ±3 and vertical asymptotes at x = ±2, so check signs on the intervals:

- x < −3 (e.g. x = −4): f < 0.
- (−3, −2) (e.g. x = −2.5): f > 0.
- (−2, 2) (e.g. x = 0): f < 0.
- (2, 3) (e.g. x = 2.5): f > 0.
- x > 3 (e.g. x = 4): f < 0.

Thus f(x) is positive on (−3, −2) and (2, 3).