To me it always seemed pretty natural. \frac{1}{f(x)} would be (f(x))^{-1}, by analogue to x^{-1}+1 vs (x+1)^{-1}. I would have been very surprised if raising just the 'f' part of the function application expression to some power were to mean raising the whole expression to that power.

I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).