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I was thinking whether f(x)=1/(1/x) and g(x)=x are the same function or not.

It seems like f(x) is essentially g(x) but according to some kind of definition:

" The composition of two functions f and g is the function h = f ◦ g defined by h(x) = f(g(x)), for all x in the domain of g such that g(x) is in the domain of f. "

Then if I take p(x)=q(x)=1/x (just to avoid confusion), then the composite function is f(x)=p(q(x)), for all x in the domain of q such that q(x) is in the domain of p(x). While in this case, q(x) or the range of q(x), namely (-∞,0)∪(0,∞), is always in the domain of p(x), and all x in the domain of q is also (-∞,0)∪(0,∞), then should the domain of f(x) be (-∞,0)∪(0,∞)? If so, f(x) and g(x) have different domains (the domain of g(x) is R obviously). Does it mean f(x) and g(x) are different functions?

(Of course I could have used just p(x)=1/x and said f(x)=p(p(x)), but it's just for avoiding some confusion.

Hope you will get that idea.)