>>> If frame S' is moving in the +X direction relative to frame S, at
>>> velocity v, and if g = 1/sqrt(1-v^2), and (0,0) -> (0',0'), and c=1,
>>> then
>>>
>>>  t' = g * (t - vx)
>>>  x' = g * (-vt + x)
>>>
>>> Inverse transform is
>>>
>>>  t = g * (t' + vx')
>>>  x = g * (vt' + x')
>>>
>>> Compose them and see what you get.
>
> You made no comment.  Did that mean you agree that these are inverses?

It depends on the operator. We are discussing Roberts use of inverse.
As you know, the set R and the operator ' + ' has an inverse ' - ' with the
identity 0.
We can transform x to x' with x' = x +1 and back again with x = x'-1, which
is analogous to what you are doing.
For the same set R and the operator '*', we can transform x to x' with x' =
2x and back again with x = x'/2.
I consider
   x' = g * (t-vx)
to have an inverse
   x = (t-x'/g) / v.

 See also

https://home.deds.nl/~dvdm/dirk/Physics/Fumbles/SetSolve.html