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