$(y-\sqrt{x})(y-\sqrt{-x})=0$

Question

$(y-x)(y+x)=0$ is a graph that combines $y-x=0$ and $y+x=0$.

I understood that by multiplication 2 functions (or more) that in the form of $a=0$ and $b=0$, so $a\cdot b=0$ will be the 2 functions. (Please explain it to me more, and why it happens).

So my question is why: $(y-\sqrt{x})(y-\sqrt{-x})=0$ is not a graph that includes $y=\sqrt{x}$ and $y=\sqrt{-x}$ How can I get to a $y$ and $x$ form which will give me a graph that includes $y=\sqrt{x}$ and $y=\sqrt{-x}$ ?

*Help me wil the correct tags

Thanks

Answer 1

Because$\sqrt x$ and $\sqrt{-x}$ are not both real numbers except for $0$. You can use $y=\sqrt{|x|}$

Answer 2

I'm only guessing what you want: Probably, you want to graph the $f:[0, \infty) \to \mathbb{R}$ and $g:(-\infty,0] \to \mathbb{R}$ of the functions $f(x) = \sqrt{x}$ and $g(x) = \sqrt{-x}$. To define a function that kind of "graphs" both functions, you can define $$ h : \mathbb{R} \to \mathbb{R}, \ h(x) = \begin{cases} \sqrt{x} \quad &\text{ if } x \geq 0, \\ \sqrt{-x} &\text{ if } x <0. \end{cases} $$