First, starting with x0 > 1, we see Newton's method
generates a sequence of points decreasing
to the root x = 1. Certainly, the corresponding picture will
be valid for any x0 > 1, so the basin of attraction of
x = 1 includes all x0 > 1.
Consequently,
the basin of attraction of x = 1 is all x0 > 0,
and the basin of attraction of x = -1 is all x0 < 0.