I am working on an exploration which starts with the following affirmation:
In this section you studied the Binomial theorem. Recall function composition from earlier in the course. In this context (in working with a function under the operation of a composition) when we raise a function to a power like $f^2$, this means $(f \circ f)(x)$. In other words, we apply the composition twice.
After this affirmation, the exploration asks a few questions relating compositions with binomial expansions.
My question is: isn't it wrong to state that raising a function to a given power is the same as applying a composition that number of times? A simple counter example would be $f(x) = 2x$.
This invalidates the whole analysis.
Also, does this make the following question not relevant/meaningful? How could I go about approaching this problem? (assumning the question really means composition)
"Will binomial expansion work for function composition? Why or why not? Use your results to make a conjecture about the binomial theorem."
(given that we are actually not raising the function to a given power, the question seem off, but of course I could be wrong)