In this list, the functions described in (i), (ii), (iv), and (vi) are not multiplicative. For example, for (i) we have
but
For (vi) we have
but
On the other hand, the function On the other hand, the square and square root functions are both multiplicative. In fact, any power function f with a description of the form
is multiplicative. To see this we calculate
Again, multiplicative functions would seem to be rather rare.