Let $z = f(x)$ be a function from $[0, \infty)$ into $\mathbb{R}$ living in the $xz$-plane, for example. Create a surface $z = g(x,y)$ by rotating the graph of $f$ about the $z$-axis.
In other words, the surface is such that $g(x,mx) = f(x)$ for all $m \in \mathbb{R}$ and all $x \in [0, \infty)$. Is there a general method for defining $g$ explicitly given $f$?
Yes in this case you have a function $z = f(x)$ in this case to obtain $g$ by revolving a function around the $z$-axis that lives in the $xz$-plane, replace $x$ by $\pm\sqrt{x^2+y^2}$ So you would have $z-f(x) = 0 \implies z-f(\pm\sqrt{x^2+y^2})=0$. Since you are on the positive axis simply use $\sqrt{x^2+y^2}$.