$\begin{array}{l}\text{ If }R=\left\{(x,y):x,y\in Z,x^2+3y^2\le 8\right\}\text{ is a relation on the set of integers }\mathrm{Z},\text{ then the domain of }\\ {\mathrm{R}}^{-1}\text{ is: }\end{array}$

$\text{ The given relation can be rewrite as }{x}^2\le 8-3y^2$

$\text{ The possible values of }y\text{ are }\{1,-1,0\}$

$\text{ Hence, the range of the relation }R\text{ is }\{1,-1,0\}$

$\text{ Therefore, the domain of the inverse relation }{R}^{-1}\text{ is }\{1,-1,0\}$