The given function are ,
f(x,y)=2xy and g(x,y)=x2+y2f(x,y)=2xy \ and \ g(x,y)=x^2+y^2f(x,y)=2xy and g(x,y)=x2+y2
We have to check whether, R2\R^2R2 is a domain of fg\frac{f}{g}gf Or not?
Clearly,
f:R2→Rf:\R^2\rightarrow \Rf:R2→R and g:R2→Rg:\R^2\rightarrow \Rg:R2→R and g(x,y)=0g(x,y)=0g(x,y)=0 ⟺ \iff⟺ x=y=0x=y=0x=y=0
Hence ,Domain of fg\frac{f}{g}gf is R2−(0,0)\R^2 -(0,0)R2−(0,0) .
Hence the statement is false.
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