Consider the function for
. Prove that this function has an inverse, determine the domain of this inverse, and find a formula to compute the inverse
.
Proof. From the discussion on page 146 of Apostol we know that a function which is continuous and strictly monotonic on an interval has an inverse on
. The function
is continuous and strictly monotonic on the negative real axis; therefore, it has an inverse. We know it is continuous since the log function is continuous on the positive real axis, and
for all
, in particular, for all
. Furthermore, we know it is strictly monotonic since
Therefore, has an inverse for all
. The domain of this inverse is the range of
which is all of
To find a formula for the inverse we set
Therefore, valid for all
.
A sketch for the graph of is given by