Say we have $f(x,y)=x^8+3x^4y^3+y^8x^{20}+y$. I have to show that for every $x\in\Bbb{R}$ there exists a unique solution to the equation $f(x,y)=0$.
So if I understand the question correctly, I take a certain $x$ and proof that there exists a unique $y$ s.t $f(x,y)=0$. So my first thought was to use the implicit function theorem, since $f_y(x_0,y_0)\neq0$ for every $(x_0,y_0)\in\mathbb{R^2}$ and $f(x,y)\in C^{\infty}$, so we're promised that for each $x$ there exists an environment $U(x)$ s.t for every $x\in U(x), f(x,y)=0\iff y=y(x)$, which is exactly saying there exists a unique $y$ s.t $f(x,y)=0$. Is my approach correct?