<s>[INST] What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor f(x) \rfloor $? I have a very basic question: What is the difference between $\lfloor f \rfloor (x)$ and $\lfloor{f(x)}\rfloor$? Are $\lfloor f \rfloor (x)$ and $f(\lfloor x \rfloor)$ equivalent?
 [/INST] You seem to be asking two different questions. $\lfloor f \rfloor (x)$ is defined as $\lfloor f(x) \rfloor$. Hence, it is indeed true that $\lfloor f \rfloor (x) = \lfloor f(x) \rfloor$.
However, $\lfloor f(x) \rfloor$ is different from $f(\lfloor x \rfloor)$. For instance, consider $f(x) = x^2$. Then $\lfloor f(x) \rfloor = \lfloor x^2 \rfloor$, while $f(\lfloor x \rfloor) = \lfloor x \rfloor ^2$. These two are not equal. For instance, choosing $x = 2.5$, we get that
$$\lfloor f(2.5) \rfloor = \lfloor 2.5^2 \rfloor= \lfloor 6.25 \rfloor = 6$$
whereas $$f(\lfloor 2.5 \rfloor) = f(2) = 2^2 = 4$$
Hence, in general
$$\lfloor f(x) \rfloor \neq f(\lfloor x \rfloor)$$
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