Answer
The domain: $D = \left\{ {\left( {x,y,z} \right)|x \ne 0,y + z \ge 0} \right\}$
The range: $\left( { - \infty ,\infty } \right)$.
Work Step by Step
The function $f\left( {x,y,z} \right) = x\sqrt {y + z} {{\rm{e}}^{z/x}}$ is defined if $y + z \ge 0$ and $x \ne 0$. Thus, the domain consists of all points $\left( {x,y,z} \right)$ lying above and including the plane $y+z=0$ and $x \ne 0$. The domain description is
$D = \left\{ {\left( {x,y,z} \right)|x \ne 0,y + z \ge 0} \right\}$
The range takes on all values in the infinite interval $\left( { - \infty ,\infty } \right)$.
