y=-x2+2x-3 can be written:
y=-3-(x2-2x+1-1), y=-3-((x-1)2-1)=-3+1-(x-1)2=-2-(x-1)2. (A parabola (inverted U-shape) with vertex at (1,-2).)
A square has a least value of zero so y cannot be more than -2. This helps us to define the range.
For all other values of x, y will be less than -2. So the range is (-∞,-2] (semi-infinite). The interval is shown in set builder notation. This can be written y∈(-∞,-2] for x∈(-∞,∞).