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