Answer
Domain: $x: (-\infty,\infty); y: (-\infty,\infty); z: (-\infty,\infty); $
Range: $f(x,y,z): (-\infty,\infty); $
Level surfaces: $f(x,y,z)=c$ (constant)
Work Step by Step
Function: $f(x,y,z)=x^{2}+4y^{2}+9z^{2}$
Domain: x, y , and z can be any real numbers;
Range: $f(x,y,z)\geq0$
Level surfaces: $f(x,y,z)=c$ (constant)
$x^{2}+4y^{2}+9z^{2}=c$ represents a series of ellipsoids.
For example, when $c=9$,
$x^{2}+4y^{2}+9z^{2}=9$ is an ellipsoid as shown in the figure.
