Which of the following represents the curve of $\left|e^{-\left|x\right|}\right|$

If $x, y$ are real numbers and function $g\left(x\right)$ satisfies $\frac{\left(g\right(x+y)+ g(x-y\left)\right) }{2}=g\left(x\right)g\left(y\right)$ and $g\left(0\right)$ is a positive real number then which of the following options may represent graph of $g\left(x\right)$?

Find the domain of the definition of the function $y=1/{(x=|x\left|\right)}^{1/2}$.