On the Wikipedia statement for the Implicit Function Theorem (IFT), part of the conclusion is the existence of a function $g(\vec x)$. Is this $g(\vec x)$ not an explicit function for $\vec y$ of $\vec x$?
I am also confused by the statement in the First Example of the same article "The purpose of the implicit function theorem is to tell us the existence of functions like $g_1(x)$ and $g_2(x)$, even in situations where we cannot write down explicit formulas." But if $g_1(x)$ and $g_2(x)$ are explicit functions, why can't we write them down?