Finding the Domain of a Function of Three Variables

Find the domain of the function \(w=f(x,y,z)=\sqrt{x^{2}+y^{2}+z^{2}-1}\) and graph the domain.

Solution  Since the expression under the radical must be nonnegative, the domain of \(f\) consists of all points for which \( x^{2}+y^{2}+z^{2}-1\geq 0.\) The domain is the set of all points on and outside of the unit sphere \(x^{2}+y^{2}+z^{2}=1\); that is, the set \(\{ (x,y,z) \,|\,x^{2}+y^{2}+z^{2}\geq 1\} \). See Figure 5.