Answer
$g(g(x))=x,\qquad x\neq 0$
Domain: $\{x|x\in \mathbb{R},\ x\neq 0\}$
Work Step by Step
$g(g(x))=-2\cdot[g(x]^{-1}$
$=-2(-2x^{-1})^{-1}$
$=-2\displaystyle \cdot\frac{1}{-2x^{-1}},\qquad x\neq 0$
$=x,\qquad x\neq 0$
... defined for all real numbers except $x=0$
