Answer
$\mathrm{Domain:}\ \ (-\infty,\infty)$
$\mathrm{See\:the\:graph\:below.}$
Work Step by Step
$\mathrm{Remember:}$ The domain of a polynomial is the entire set of real numbers.
The limiting factor on the domain for a rational function is the denominator, which cannot be equal to zero.
$\mathrm{Domain:}\ \ (-\infty,\infty)$
To graph the parabola, we need at least three points. First of all, find the vetex of the parabola by using the formula for the $\ \mathrm{x-coordinate}\ $ as:
$x=\frac{-b}{2a}$
$f(x)=y=-x^2-2x+1$
Here, $\ a=-1,\quad b=-2,\quad and\quad c=1$
$x=\frac{-(-2)}{2(-1)}=\frac{2}{-2}=-1$
Put this $\ \mathrm{x-coordinate}\ $ in the function to get the corresponding $\ \mathrm{y-coordinate}.$
$y=-(1)^2-2(1)+1=-2$
$\mathrm{Vertex:}\ (-1,2)$
Now apply the quadratic formula to find the roots as:
$x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$x_{1,\:2}=\frac{-(-2)\pm \sqrt{(-2)^2-4(-1)1}}{2(-1)}$
$x_1=-1-\sqrt{2},\quad x_2=\sqrt{2}-1$
So, the other two points are, $\ (-1+\sqrt{2},0) \ \ and \ \ (-1-\sqrt{2},0)$
See the graph below.
