I have recently taken great interest in studying the properties of the function [tex] f(x) = x^x [/tex], and I was wondering: is there any way to prove whether [tex] f(x) = x^x [/tex] is an injective (i.e. one-to-one) function? I realize that if I can prove that if the inverse of [tex] f(x) = x^x [/tex] is also a function, then [tex] f(x) = x^x [/tex] is injective. The problem is: [tex] f^{-1}(x) [/tex] is non-algebraic, so I can't figure out whether it's a function or not. Does anyone know another way to prove whether or not [tex] f(x) = x^x [/tex] is injective?(adsbygoogle = window.adsbygoogle || []).push({});

NOTE: The domain of this function is real numbers.

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# Is x^x injective?

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