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

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

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

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