is there any proof or theorem to say that if the inverse function $ y=f^{-1}(x) $ is POSITIVE in the sense $ f^{-1}(x) >0 $ for $x \ge 0 $ then the function $ f(x) \ge 0 $ will be also positive on the interval $ (0, \infty) $ ? in the sense that $ f(f^{-1}(x))=x $
so if a function IS POSITIVE its inverse will be also positive :D on the same interval
since taking the inverse function may be considered as reflection of the function $ y=f(x) $ across the line $ y=x $ i think that my aseveration is true.