Originally Posted by

**emakarov** Yes. If we just had increasing functions on (1, 5) whose ranges were (0, 2) and (0, 4), then the range of the product would be (0, 8). Here we have non-monotonic functions and one of them: |x - 3| has a zero in 2. So, their product would involve a 0, unlike (0, 8). However, |x - 1| is viewed over (1, 4) excluding 3 because the the restriction 0 < |x - 3|. So, |x - 1| |x - 3| is never 0 and their range is (0, 8) after all (so the range of 2|x - 1| |x - 3| is (0, 16)).

All I am saying, is the by multiplying the ends of the interval ranges of two functions, one only gets the some superset of the range of the product, which may be a nice approximation to the actual range.