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While studying inverse trigonometric functions ,a thought struck me that inverse function like $sin^{-1} {x}$ and $cos^{-1} {x}$ have domain [-1,1] but what about rest of the Real space/Real line

Can't we have a function that in a certain domain be inverse of some function and outside that certain domain(or to some extent) be the inverse of another function/functions?

Or atleast be inverse of n = 2 functions?

Is there any example of any of them?

If so what are the generalized properties of such functions?(their derivatives,behaviours,etc...)

Where can I look for more information?

ANy well known applications of such functions outside mathematical research say in Physics/Engineering? and why they are incorporated as such?(what property makes the feasible for such branches)

I s there any property/axiom/theorem barring to do so?(construct such f(x)s)

IS there any branch of maths exist that especially analyses this kind of function?

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  • $\begingroup$ a function that ... outside that certain domain A function is always defined on a certain domain. It's not even defined outside it, let alone be the inverse of some other function (or not). $\endgroup$ – dxiv Oct 24 '16 at 6:15
  • $\begingroup$ I meant to say a function with a very wide domain(By wide I meant domain that encompasses the domain of inverse functions of some n functions) and behave as inverser of some function in that domain say in [-1,1] behave as sin^-1(x) and outside that behave as inverse of some other function function f(x) @dxiv $\endgroup$ – Xasel Oct 24 '16 at 6:18
  • $\begingroup$ Please review your definition of a function. If it has an inverse at all, then that inverse is unique, and the inverse of the inverse is the original function, so the whole inverse of n functions makes no sense. $\endgroup$ – dxiv Oct 24 '16 at 6:22
  • $\begingroup$ Sorry i don't know whether inverse has to be unique or not can you gave pointers to link or such or a theorem that proves that a functions has unique inverse(because I think we can define a function that in the some interval is stricted to be inverse of some function f_1(x) but can behave as whatever we wished it to behave(i mean constructed as such) outside that interval) @dxiv $\endgroup$ – Xasel Oct 24 '16 at 6:28
  • $\begingroup$ Uniqueness follows straight from the definition. For the definition, see for example Inverse function. $\endgroup$ – dxiv Oct 24 '16 at 6:32
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Let $f_i\ (i\in I)$ be functions with domain $D_i\ (i \in I)$ for a set $I$, s.t. the sets $D_i$ are pairwise disjoint. Then you can construct a function $f$ s.t. $f|_{D_i} = f_i\ (i\in I)$. If $f$ is bijective, an inverse function $f^{-1}$ exists.

It then holds that $f^{-1}|_{f_i(D_i)}=f_i^{-1}$.

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