yes, the domain is a subset of . but in this case, if f(x,y) makes sense for all real numbers x and y,
then the domain equals .
now is always positive, so we know the range is a subset of the positive reals. the question is, can
turn out to be any real number? and it can: let's say we have a certain real number r in mind. then we could pick x = 0, and y = r.
so now we ask, given that can be any real number, can be any positive real number?
suppose s is positive. then ln(s) makes sense, so we can pick (x,y) = (0,ln(s)), and then .
so we see any positive real number is in the range of f, and only positive real numbers are in the range of f.
so range(f) = . this is sometimes also written: or as the interval