show that $\mathcal{A}_{f}$ is a $\sigma$-algebra

Question

Given:
S is set and $f: S \to \mathbb{R}$

Problem:
Showing that $\mathcal{A}_f = \{ f^{-1}(B): B \in \mathcal{B}(\mathbb{R}) \}$ is a $\sigma $ - algebra.

My approach:
Try to show all three properties of a $\sigma $ - algebra:
So I would like to show that the following three statements hold:

1. $\emptyset \in \mathcal{A}_f $ and $S \in \mathcal{A}_f $
2. $\forall A \in \mathcal{A}_f: A^{c} \in \mathcal{A}_f $
3. $A_1,A_2,.... \in \mathcal{A}_f \ \ \cup_{n=1}^{\infty} A_n \in \mathcal{A}_f$

I guess I should do something with measurability of a $f$, but I can't figure out what, since I don't know nothing about the measurability of f.

Question:
Can someone tell me how I should tackle this problem? How can you for example show property (1) in the first place?