The function is defined for all real #x#, so the domain is just the real numbers, #RR#.
To find the range, we will consider the function as a parabola. Since the #x^2# term is negative, we know the parabola will be concave down (#nn#). Therefor the range will be between #-oo# and the vertex of the parabola.
The x coordinate of the vertex of a parabola #ax^2+bx+c# can be found using the following formula:
#-b/(2a)#
Plugging in our numbers, we get:
#(-(-3))/(-2)=-3/2#
Now, we want the #y# coordinate of the vertex, so we plug in the value into the original function:
#-(-3/2)^2-3(-3/2)-3=-9/4+9/2-3=-9/4+18/4-12/4#
#=-3/4#
So we know the vertex is at #-3/4#. This means that the range of the function is #{y|y<=-3/4}#
