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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.

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    I think there is a bijective constraint, as you say nothing about $x<0$ behavior2012-10-03
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    thanks Bernhard.. however if we assume $ f(x)=f(-x) $ i think i this case we could always take the positive branch of the function2012-10-03
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    Your affirmation concerning the reflection is indeed sufficient (provided there is an inverse of course).2012-10-03
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    you can always DRAW an inverse for any function at least for piecewise continous function, so there is always an inverse2012-10-03
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    My comment was aimed at the interval $(0,\infty)$2012-10-03

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