Let $f(x)\in \mathbb{F}_q[x]$ and let $X$ be the curve over $\mathbb{F}_q[x]$ defined by $\psi(x,y)=(f(x)-f(y))/(x-y)$. I want to show that if $(a,b)$ is a simple rational point of $X$ then the absolutely irreducible component of $X$ containing $(a,b)$ is defined over $\mathbb{F}_q$.
I tried to show it by contradiction. How can we say that $(a,b)$ belongs at least two component of $X$ then $(a,b)$ is singular? Which argument can we use here? I would be grateful for any help.