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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?

enter image description here

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    $\begingroup$ I think $m$ and $q$ are backwards when you composed them... $\endgroup$
    – abiessu
    Apr 11, 2020 at 13:37
  • $\begingroup$ You are right! Thanks for pointing this mistake. $\endgroup$
    – user654868
    Apr 11, 2020 at 13:42

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