Originally Posted by

**tttcomrader** I was reading a problem and I saw this:

Let f and g be functions with range of $\displaystyle \mathbb {R} \cup \{ \infty \} $

Then $\displaystyle (fg)^{-1} ( \{ \infty \} ) = [ f^{-1}( \{ \infty \} \cap g^{-1} ([ - \infty , 0))] \cup [ f^{-1} ( \{ - \infty \} ) \cap g^{-1}(0, \infty ] ] $$\displaystyle \cup [ f^{-1} [ - \infty ,0) \cap g^{-1} ( \{ \infty \} ) ] \cup [ f ^{-1}(0, \infty ] \cap g^{-1} ( \{ - \infty \} ) ]$

Why is this true? I'm really confuse about that.

Thank you.