I was thinking about this recent question : what is the range of $f(x)= \frac {1} {\sqrt{x^2 - 1}}$.
I tried to express this function as a function composition , that is,
with
$i(x)=\frac 1x$
$s(x)= \sqrt x$
$m(x)= x-1$
$q(x)= x^2$
as : $f(x)= i(s(m(q(x))))$.
At this point I realized that it was difficult to express symbolically the domain and the range of this function composition.
If I am correct, when the composition has only two components, one can say that :
dom $( f\circ g) = \{x| x\in$dom$(g) \land g(x)\in$dom$(f)\}$
ran$ ( f\circ g) = f[$ran$(g) \cap $dom$(f)]$.
My question is : how to express the same thing with more than 2 component functions? possibly, with $n$ components?