Explicit function of a cylinder

Question

Can I transform an implicit function of a cylinder to explicit form?

Lets say we have a cylinder

$$x^2 + y^2 - 1 = 0$$

and we want to have it expressed as function $f(x,y,z)$.

Answer 1

If what you want is an equation of the form $z=f(x,y)$, forget it: such a function would, by definition, give a unique value of $z$ for every pair of values $(x,y)$, but on your cylinder, a single pair $(x,y)$ may correspond to infinitely many values of $z$. That is, what would you want $f(1,0)$ to be? If what you want is $f(x,y,z)=0$, then, as lhf writes in the comments, you already have it. Think of it as $x^2+y^2+0z-1=0$, if it helps.