Notation Question with regard to functions

Question

Let $f : N → N$

Let $E(f)$ be the function defined by $E(f)(n) = 2^{f(n)}$.

Does $E(f)(n)$ mean $E(f(n))$? or $E(f)(n)$?

Answer 1

$E(f)$ is the function that when applied to $n$ returns $2^{f(n)}$. In other words, the operator $E$ takes in $f$ and returns another function from $N$ to $N$.

Thus, strictly speaking, the expression $E(f(n))$ makes no sense because $f(n)$ is a number and the input for $E$ is a function. On the other hand, we can be loose in our interpretation. For example, $E^5(n^2)$ can be interpreted as: let $f(n)=n^2$, then $E^5(n^2)=E[E[E[E[E(f)]]]]$; at each step, $E$ is applied to a function.