Counting circuits with constraints

Please forgive me if this question is trivial, I couldn’t come up with an answer (nor finding one).

In order to show that there are boolean functions $f : {0,1}^n rightarrow {0,1}$ which can be computed only using circuits of size $Omega(2^n/n)$, we use a counting argument: there are at most $O(2^{k log k})$ circuits of size $k$, and $2^{2^n}$ such functions.

Suppose that I am interested in counting circuits of size $k$ that compute different functions. The "simple" counting argument won’t work since it may be possible that two "syntactically" different circuits actually compute the same function.
In other words, I want to bound the size of the set: $$F = { f: {0,1}^n rightarrow {0,1} | f text{ can be computed using a circuit of size }k } $$

Then $|F| < $ the number of circuits of size $k$ (since any circuit computes one function), but how can I bound $|F|$ from below? (i.e. $ x <|F|$)