I am currently working on an exercise where I am considering the function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ defined by
$f(x,y,z) = (x^2+y^2-4)^2 + z^2$
The exercise asks you to first find the regular values of $f$. Then, it asks you to identify the different types of surfaces we obtain from $f^{-1}(a)$, where $a$ is a regular value.
I found the critical points of $f$ to be $(x,y,z) = (0,0,0)$ and the set of all $(x,y,z)$ such that $z = 0$ and $x^2+y^2=4$. Since $f(0,0,0) = 16$ and $f(x,y,z) = 0$ for all $z = 0$ and $x^2+y^2=4$, I concluded that the regular values of $f$ are all real numbers except $0$ and $16$. Now, I do not know how to proceed in identifying the different types of surfaces we obtain from $f^{-1}(a)$ for all of these regular values.
Any help would be greatly appreciated.