Question

must a function that is decreasing over a given interval always be negative over that same interval? explain.

Explanation:

A function decreasing over an interval means that as the input (x - values) increases within that interval, the output (y - values) decreases. However, the sign (positive or negative) of the function values in that interval is a separate concept. For example, consider the function \(y=-x + 5\) over the interval \((0,3)\). The function is decreasing (since the slope \(m=- 1<0\)), but \(y(0)=5\) (positive) and \(y(3)=2\) (positive). Another example: \(y = -x-1\) over \((0,3)\) is decreasing and \(y(0)=-1\) (negative) and \(y(3)=-4\) (negative). So, the decrease in the function (related to the rate of change) is about the trend of the function values as \(x\) changes, not about whether the function values themselves are positive or negative.

Answer:

No, a function that is decreasing over a given interval is not always negative over that same interval.