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|$ )