Can we apply a function to a set? What does $f(A)$ mean when $A$ is a set of numbers? For eg., let  $ A = \{-2, -1, 0, 1, 2 ,3\} $
Let $f(x) = \lfloor \frac {x^2}{3}\rfloor $
What is $ f(A) $ ?
In case anyone is wondering, this is homework, but I am not sure how to proceed as I can't find anything online regarding this.
 A: Yes, we can, but it's an abuse of notation. To be really strict (though only a set theorist or logician is likely to be this strict, hence why Peter answered in the negative!), you'd write $f"A$ or $f^{\to}(A)$, but it's clear from context what $f(A)$ is supposed to mean: the image of everything in $A$ under $f$.
In your case, you want $$f"A = \left\{\left\lfloor \frac{x^2}{3} \right\rfloor: x \in A\right\}$$
which is $\{0,1,3\}$.
A: $f$ has been defined as a function on numbers (of some kind). Its domain and codomain comprise numbers (not sets of numbers). 
So $f$ acts on numbers, not sets. We can't, so to speak "pass a set through" this function. Without further explanation, $f(A)$ makes no sense, and the question "what is $f(A)$?" is simply ill-defined. 
But as Patrick Stevens points out, there is a common bad(?) habit of using something like '$f(A)$' to mean $\{f(x) \colon x \in A\}$. Note however that with the notation understood this way, when we evaluate $f(A)$ we are still not "passing the set [of numbers] through the function", we are passing numbers (members of the set $A$) to the function!  
That use of '$f(A)$' can however lead to tangles in some situations, and since we have alternative unambiguous notations for the same thing (the simple one I like is '$f[A]$'), it is best avoided. But if the slang(?!) usage is intended, as is probably the case, then Patrick gives the answer!
