Constructing inverse function that is inverse of n functions?

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...)

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?

• 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). – dxiv Oct 24 '16 at 6:15
• 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 – Xasel Oct 24 '16 at 6:18
• 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. – dxiv Oct 24 '16 at 6:22
• 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 – Xasel Oct 24 '16 at 6:28
• Uniqueness follows straight from the definition. For the definition, see for example Inverse function. – dxiv Oct 24 '16 at 6:32

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 $f^{-1}|_{f_i(D_i)}=f_i^{-1}$.