Question

Must a function that is decreasing over a given interval always be negative over that same interval? Explain.

Answer 1

No.

Explanation:

Firstly, observe the function `f(x)= -2^ x`
Clearly, this function is decreasing and negative (i.e. below the x-axis) over its domain.

At the same time, consider the function `h(x)=1-x^2` over the interval `0<=x<=1`. This function is decreasing over the said interval. However, it is not negative.

Therefore, a function need not be negative over the interval it is decreasing on.