Must a function this is decreasing over a given interval always be negative over the same interval? Explain

It is certainly possible for a function decreasing over a certain interval to be negative, but no rule that says it must be. On the other hand, where the function is decreasing, the rate of change of the function must be negative.

For a function to be decreasing over an interval, the outputs on the function are getting smaller as the inputs of the function are getting larger, but the outputs could be positive or negative. For a function to be negative over an interval, the outputs must be negative, while the inputs could be positive or negative.