I was joking with someone when I spelled some proper noun in a mathematical equation:
$$\tan x \in f(u)$$
Obviously this function needs to have a value that is a set in order to make sense. Hence, this got me thinking: can a function have sets as argument or value?

For example maybe there could be a function so that $f(x)=\mathbb{R}^x$, or perhaps $f(\mathbb{R}^x) = x$ or even a function that adds one to each of the elements of the argument set $f([0,\infty)) = [1,\infty)$.

Can such functions exist?