Applying the Implicit Function Theorem to the Unit Sphere

Question

The unit sphere S given by $x^2+y^2+z^2=1$ intersects each of the three axis at 2 points, at these points, what variables can be solved for? For example, S intersects the x-axis at $(\pm1,0,0)$, I can come up with functions $f_{1}(y,z)=\sqrt{1-y^2-z^2}$ and $f_{2}(y,z)=-\sqrt{1-y^2-z^2}$, but I don't really understand the connection between these and the Implicit Function Theorem.

Answer 1

Let $F : \Bbb{R}^3 \to \Bbb{R}$ be given by $F(x,y,z) = x^2 + y^2 + z^2 - 1$. Then the set on which $F = 0$ (i.e., $F^{-1}(\{0\})$) is exactly the unit sphere $S$. If you look at $\frac{\partial F}{\partial x}$ evaluated at the point $(1,0,0) \in S$, you find that it is non-zero. The implicit function then says that there is some function $g(y,z)$ such that for $(x,y,z) \in S$ sufficiently close to $(1,0,0)$, you have $F(g(y,z),y,z) = 0$. What is $g$? It is exactly the $f_1$ you posted.