Is x^x injective?

[Amherstclane]

I have recently taken great interest in studying the properties of the function f(x) = x^x , and I was wondering: is there any way to prove whether f(x) = x^x is an injective (i.e. one-to-one) function? I realize that if I can prove that if the inverse of f(x) = x^x is also a function, then f(x) = x^x is injective. The problem is: f^{-1}(x) 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 f(x) = x^x is injective?

NOTE: The domain of this function is real numbers.