In this question, all operations are defined over a finite field of prime order $\mathbb{F}_p$, and all polynomials are defined over $\mathbb{F}_p[x]$. All values and polynomials are non-zero. By $b^{-1}$ we mean the multiplicative inverse of $b$.
Let $F(x)$ be a polynomial of degree $d$ and $x=[x_1,...,x_{d+1}]$. Let $y_{i}=F(x_i)$.
Consider the case where we interpolate a polynomial $G(x)$ using $d+1$ pairs $(x_{i},y_{i}^{-1} )$.
Question: Is there any relation between $F(x)$ and $G(x)$?