Answer
Domain: x and y can be any real numbers in the x-y plane.
Range: $f(x,y) \geq 0$
Level curves: $9x^{2}+y^{2} = c$ (ellipses).
Work Step by Step
Function: $f(x,y)=9x^{2}+y^{2}$
Domain: x and y can be any real numbers in the x-y plane.
Range: $f(x,y) \geq 0$
Level curves: $f(x,y) = c$ (constant), i.e. $9x^{2}+y^{2} = c$ which are a series of ellipses.
E.G. for $c=36, 9x^{2}+y^{2} = 36$, can be plotted as an ellipse as shown in the figure.
$x^{2}/4+y^{2}/36 = 1$
