Solution: Since $ \sin^2 x + \cos^2 x = 1 $, we write $ f(x) = \frac1\sin^2 x + 4\cos^2 x = \frac11 + 3\cos^2 x $. Let $ y = \cos^2 x $, so $ 0 \leq y \leq 1 $. Then $ f(x) = \frac11 + 3y $. As $ y $ varies from 0 to 1, $ 1 + 3y $ varies from 1 to 4, so $ f(x) $ varies from $ \frac14 $ to $ 1 $. Hence, the range of $ f(x) $ is $ \left[\frac14, 1\right] $. The final answer is $ \boxed\left[\frac14, 1\right] $. - Abu Waleed Tea