Answer
$$D = \left\{ {\left( {x,y} \right):xy \geqslant 0,{\text{ }}\left( {x,y} \right) \ne \left( {0,0} \right)} \right\}$$
Work Step by Step
$$\eqalign{
& g\left( {x,y} \right) = \sqrt {\frac{{xy}}{{{x^2} + {y^2}}}} \cr
& {\text{Because }}g{\text{ involves a square root}},{\text{ its domain consists of }} \cr
& {\text{ordered pairs }}\left( {x,y} \right){\text{ for which }}\frac{{xy}}{{{x^2} + {y^2}}} \geqslant 0,{\text{ }} \cr
& {x^2} + {y^2}{\text{ is always positive, then }}xy \geqslant 0 \cr
& {\text{besides }}{x^2} + {y^2} \ne 0 \cr
& {\text{Then,}} \cr
& xy \geqslant 0{\text{ and }}{x^2} + {y^2} \ne 0 \cr
& xy \geqslant 0{\text{ and }}\left( {x,y} \right) \ne \left( {0,0} \right) \cr
& {\text{Therefore, the domain of }}h{\text{ is }} \cr
& D = \left\{ {\left( {x,y} \right):xy \geqslant 0,{\text{ }}\left( {x,y} \right) \ne \left( {0,0} \right)} \right\} \cr} $$
