Qualitatively describing level surfaces from equations alone Describe level surfaces: 
  (i) $f(x,y,z)=z-x^2-z^2$ (ii) $f(x,y,z)= 3x-y+4z-1$


Not sure think (i) is a type of paraboloid.
Think (ii) is a plane not sure what else I can describe in both cases any thoughts would be appreciated.
The first family of level sets are NOT paraboloids. 

Here is a simple way to obtain them:

$z-x^2-z^2=k$ or  $z^2 -z+x^2=-k$ can be written $x^2+(z-\frac{1}{2})^2-\frac{1}{4}=-k$ of the form $Z^2+X^2=a$ with $a=0.25-k$. 

Thus there are 2 cases, constituting a family of co-axial cylinders (if non void!):


$k \leq 0.25$; a cylinder with radius $\sqrt{0.25-k}$ and axis parallel to $y$-axis passing at $z=1/2$. The case $k=-0.25$ gives a degenerate cylinder reduced to its axis.
$k&gt;0.25$: void sets.


The second equation is much simpler : $3x-y+4z-1=k$ is the equation of a plane for any $k$. All these planes share a common normal vector $(3,-1,4)$. They are thus all parallel.
You're correct in both cases. One way to see why is to set $f(x,y,z)=0$ (or more generally any constant $c$) and solve for $z$.