I wonder what a function f(x) with the property that f(f(x)) = e^x might look like. (My usual caveat: DON’T TELL ME & RUIN THE WONDERING STAGE.)
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In particular, the derivative is never zero and everywhere continuous, so we also have a guarantee that f is invertible (by whatever that theorem is called that says a function is locally invertible if its derivative is continuous and nonzero in that neighborhood)
But doesn’t that just say that f(f(x)) is is invertible? Pardon my analysis is a bit rusty
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Yes, of course it is! f(f(x)) = e^x, which is invertible.
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Continuing along that line of thought, f is also positive-definite.
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From the expression you have above in the box, if you assume f is smooth enough and defined for all R then f' can never vanish. And then it's inverse is found by just drawing the graph and reflecting on the x=y line (I mean there are theorems, but that's even easier :) )
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Of course, the assumption that it is well-defined on all of R needs to be justified
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