root(1-root(1-root(2-x²)))

hi ,

 for  root(2-x²) to be real we must have x^2=<2 emplies  that

x ε [-root2 , root 2]

then, for (1-root(2-x²) to be real , root(2-x²)=<1 , emplies (2-x²)=<1

x^2 >= 1  so , x ε [-root2 , -1 ] U [1, root2 ]

 we need not further analysis becoz   root(1-root(2-x²)) will always less than 1

so, the domain will be     x ε [-root2 , -1 ] U [1, root2 ]

-2<=x<=-1 and 1<=x<=2