Answer
$$D = \left\{ {\left( {x,y} \right):y < {x^2}} \right\}$$
Work Step by Step
$$\eqalign{
& g\left( {x,y} \right) = \ln \left( {{x^2} - y} \right) \cr
& {\text{Because }}g{\text{ involves a natural logarith function}},{\text{its domain consists}} \cr
& {\text{of ordered pairs }}\left( {x,y} \right){\text{ for which }}{x^2} - y > 0. \cr
& {x^2} - y > 0 \cr
& {x^2} > y \cr
& or \cr
& y < {x^2} \cr
& {\text{Therefore, the domain of }}f{\text{ is }}D = \left\{ {\left( {x,y} \right):y < {x^2}} \right\} \cr} $$
