# Question about the result of the implicit function theorem

Consider $$\textbf{F}(x,y,z) = ((x+z)^2, y-z) = 0$$ Taking the Jacobian at the point $$(0,0,0)$$ gives a matrix of rank $$1$$, which from my understanding means that the implicit function theorem cannot guarantee an implicit function $$g(z)=(x,y)$$ in the neighborhood $$(0,0,0)$$.

However, the book I am reading suggests that an implicit function $$\textbf{does}$$ exist, in the form of $$g(z)=(-z,z)$$, and concludes that $$\textbf{F}(g(z),z)=(0,0)$$ in a neighborhood of $$(0,0,0)$$.

I really don't understand the intuitive idea and result here at all (and maybe I'm completely misunderstanding the implicit function theorem). I know that the implicit function theorem shows when we can write $$n-k$$ "passive variables" as a function of the one "free variable" (in this case $$z$$). I have some questions:

1. Is the implicit function theorem a sufficient condition but not a necessary condition for finding an implicit function $$g$$ of the $$k$$ free variables at a point? How do we know $$g(t)=(-z,z)$$ satisfies the conditions stated in a "neighborhood of $$(0,0,0)$$"?
2. If so, how can I reconcile these two results with each other? What is missing from the implicit function theorem?