partial inverses of x tanh(log(1 + e^x))

Question

$f(x) = x \tanh(\log(1 + e^x))$ is not one-to-one. It has a minima at $x=-1.19243$.

We can divide the domain of the function into two sets: $\mathbf{A} = \{x|x \lt -1.19243...\}$ and $\mathbf{B} = \mathbb{R} - A$. How can one find the partial inverses for the two domains?