Help analyzing inverse of an abs function

I am trying to find an expression for

And then the algebra

First solution:

Second Solution:

I think my calculator is telling me that the first solution I did is true for and and the second solution is true for all reals.

If I am understanding my calculator correctly, how do I analytically determine that the second solution is the inverse of f for all real values of x? And if this is incorrect, how do I analyze the functions I found for ?

Thanks a lot if you can help!