as you know a function f has an inverse function if f is monotonously increasing or monotonously decreasing.
So your function has 3 intervalls where f is monotonous:
So you are looking for a lower bound of the intervall which contains +infinity.
With your suggestion you would look for the upper bound.
I have to send you this post without checking my text because the preview doesn't work. So watch out: There are possibly some mistakes.