Entire function maps bounded sets to bounded sets

Question

I was doing some problems in Complex Analysis… And I came across this.

Let $f: \mathbb{C} \to \mathbb{C}$ be entire. Then for any bounded set $B$, f ($B$) is bounded.

Now I know that if an entire function is constant then the above statement is necessarily true.

What about non-constant entire function?

All I know is that a non-constant entire function is unbounded. (By Liouville’s Theorem)

I am summarising my doubts here:

$1)$ What is the image of a bounded set under this non-constant entire function?

$2)$ What can we say about its image on the unbounded set?

Thanks for your time.