  Given a polynomial $\T_n$ of degree $n$, consider the inverse image of $\R$
and $[-1,1]$, denoted by $\T_n^{-1}(\R)$ and $\T_n^{-1}([-1,1])$, respectively.
It is well known that $\T_n^{-1}(\R)$ consists of $n$ analytic Jordan arcs
moving from $\infty$ to $\infty$. In this paper, we give a necessary and
sufficient condition such that (1) $\T_n^{-1}([-1,1])$ consists of $\nu$
analytic Jordan arcs and (2) $\T_n^{-1}([-1,1])$ is connected, respectively.
