# Math Help - solution set of f(x)g(x)=0 and (f(x))^2+(g(x))^2=0 given...

1. ## solution set of f(x)g(x)=0 and (f(x))^2+(g(x))^2=0 given...

Hi, I have this problem which I don't know how to solve...I don't want the answer, but the properties I need to work it out so that I can apply to other problems (but if I need to be shown with an example, then I don't mind being shown how to solve it). Thanks a lot!

Let the solution set of $f(x)=0$ be $A$ and the solution set of $g(x)=0$ be $B$. What is the solution set of $f(x)g(x)=0$? What is the solution set of $(f(x))^2+(g(x))^2=0$? What relationship holds between the solution set $P$ of $f(x)<2$ and the solution set $Q$ of $f(x)<-1$?

2. $f(x)g(x)=0$ iff $f(x)=0$ or $g(x)=0$ iff $x\in A\cup B$.

$f(x)<-1\Rightarrow f(x)<2$. So $Q\subset P$.

3. $\left[ {f(x)} \right]^2 + \left[ {g(x)} \right]^2 = 0$ if and only if $f(x) = 0\;\& \;g(x) = 0$ together.
Therefore that solution set is $A\cap B.$