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.)
Next,