The absolute value function fails to be one-to-one on the real line (it fails the horizontal line test). Therefore, it has no inverse. Are there any artificial domain restrictions we should know about here?
If you're only interested in an inverse relation, not a function, then I would agree that the situations are analogous. Incidentally, you should write the inverse as the following:
The graph of the inverse relation is merely the original graph flipped about the line , as indeed, all inverse relations in 2 dimensions are produced in the same way.