Recall that:
[tex](f\cdot g)(x)=f(x)g(x)\text{.}[/tex]
Then, if:
[tex]\begin{gathered} x\in\text{Dom(f)}\cap Dom(g) \\ \text{then} \\ x\in\text{Dom(f}\cdot g)\text{.} \end{gathered}[/tex]
Now, notice that:
[tex]\begin{gathered} x\in\text{Dom(f}\cdot g)\text{ if } \\ f(x)\text{ and g(x) exists,} \\ \text{then} \\ x\in\text{Dom(f)}\cap Dom(g). \end{gathered}[/tex]
Therefore:
[tex]\text{Dom(f}\cdot g)=\text{Dom(f)}\cap Dom(g).[/tex]
Answer: True.
