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If I have the function f(x) = 1/(1+x), with the domain restriction x

__>__0 and

g(x) = (1-x)/x, with the domain restriction 0 < x

__<__1 ....

Then these functions are inverses. If I compose them to test this, then of course, I get the identity function in both cases....but do I need to specify a domain restriction on f(g(x)) and g(f(x))? I believe, that if I do so, then the domain of both composites should be 0 < x

__<__1. Firstly, am I correct, and secondly, what does this actually mean....in terms of the functions being inverses? If we were to graph the composites, would we just have a line segment? If the answer is yes, does this have any affect on the inverses operation upon each other?

Thanks for any help or insight anyone can give, or any other similar examples I could investigate.