1. For the function [latex]f\,(x,\ y,\ z)=\frac{3x-4y+2z}{\sqrt{9-x^{2}-y^{2}-z^{2}}}[/latex] to be defined (and be a real value), two conditions must hold:
   1. The denominator cannot be zero
   2. The radicand cannot be zero

   Combining these conditions leads to the inequality

   [latex]9-x^{2}-y^{2}-z^{2}\,>\,0[/latex].

    

   Moving the variables to the other side and reversing the inequality gives the domain as

   [latex]\text{domain}\ (f)=\{(x,\ y,\ z)\in\mathbb{R}^{3}|x^{2}+y^{2}+z^{2}\,<\,9\}[/latex],

    

   which describes a ball of radius [latex]3[/latex] centered at the origin. (Note: The surface of the ball is not included in this domain.)

2. For the function [latex]g\,(x,\ y,\ t)=\frac{\sqrt{2t-4}}{x^{2}-y^{2}}[/latex] to be defined (and be a real value), two conditions must hold:
   1. The radicand cannot be negative.
   2. The denominator cannot be zero.

   Since the radicand cannot be negative, this implies [latex]2t-4\,\geq\,0[/latex], and therefore that [latex]t\,\geq\,2[/latex]. Since the denominator cannot be zero, [latex]x^{2}-y^{2}\neq{0}[/latex], or [latex]x^{2}\neq{y}^{2}[/latex], which can be rewritten as [latex]y\neq\pm{x}[/latex], which are the equations of two lines passing through the origin. Therefore, the domain of [latex]g[/latex] is

   [latex]\text{domain}\ (g)=\{(x,\ y,\ t)|y\neq\pm{x},\,t\,\geq\,2\}[/latex].

The level surface is defined by the equation [latex]4x^{2}+9y^{2}-z^{2}=1[/latex]. This equation describes a hyperboloid of one sheet as shown in the following figure.

Figure 1. A hyperboloid of one sheet with some of its level surfaces.