Finding domain of $f \circ g$ I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not.
If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$
$ $
Then $f \circ g = x$
$ $
I think domain of $f \circ g $ is $\mathbb{R} - \left\{0\right\}$
$ $
But many ppl I know are having an opinion that domain is $\mathbb{R}$ 
$ $
Which is true, OR is it just a matter of convention.

Functions are like input output machines and composite functions are a combination of them. Thus when you consider f(g(x)), the domain cannot be greater than the domain of g(x) because if g(x) cannot process a x value then how can f(g(x))?
It does not do to just simplify composite functions because if f(g(x)) = x then can't we just make another function named h(x) =x but we don't. Why? so that the function f(g(x)) depends on the domain of the functions f(x) and g(x).
