inverse of a power series with one specific solution

Question

I have a school assignment and for now, I don't know where to start, I have to show that there exist a surrounding $U$ of $0$ where the following is true: If $y\in U$ , the equation $y=\frac{x}{f(x)}$ has one solution, $x=g(y)$. If $f(x)=\sum_{k=o}^{\infty}a_kx^k, R>0$ and $a_o\neq0$. As said, I'm quite confused about where to start.

Answer 1

Hint: if $\Gamma$ is a small circle around $0$, and $F(x,y) = y - x/f(x)$, $$ g(y) = \dfrac{1}{2\pi i} \oint_\Gamma z \dfrac{\partial F(z,y)/\partial z}{F(z,y)}\; dz$$