Let . Show that is strictly monotonic on . Find the domain of the inverse of , denoted by . Find a formula for computing for each in the domain of .
First, to show is monotonic let with . Then we consider
But, since by assumption, we have . The second term in the product is also positive since it is a sum of positive terms. Therefore,
Hence, is strictly increasing on . (Of course, there are faster ways to discover the is increasing, but we do not know yet what is a derivative.)