What does it mean to define $f^{-1}(a)$ in the context of level sets, tangent planes and normals?
E.g. I have an exercise that starts like:
Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}, f(x,y,z)=e^{x+2y}\cos(z)-xz+y$ and $S=f^{-1}(2)$, where $S$ is the level set.
So what does the $f^{-1}(2)$ actually define and how is it used in the context of level sets, tangent planes and normals, e.g. in the above exercise?
$S=f^{-1}(2)$ is the subset of $\mathbb{R}^3$ such that $$ e^{x+2y}\cos(z)-xz+y=2 \quad \mbox{for} \quad (x,y,z)\in S $$ so it is a surface in $\mathbb{R}^3$ analougous to a level curve for a function of two variables.