Some functions are increasing for all x and others are decreasing for all x.
For an increasing function an increase in x produces an increase in y so that
For a decreasing function an increase in x produces a decrease in y. so that
Other functions have intervals on which the function is increasing and intervals where it is decreasing.
If a function is increasing or decreasing it is said to be monotonic. If it is increasing or decreasing on an interval it is said to be monotonic on that interval.
is monotonically increasing for
and monotonically decreasing for