The range is the set of values of f(x) you can get from your domain. So given f(x)=x^2-1 and domain D={-2,-1,0,2}, all you have to do is plug in the elements of your domain into your function.
color(red)(x=-2)
[1]" "f(-2)=(-2)^2-1
[2]" "f(-2)=4-1
[3]" "color(red)(f(-2)=3)
color(blue)(x=-1)
[1]" "f(-1)=(-1)^2-1
[2]" "f(-1)=1-1
[3]" "color(blue)(f(-1)=0)
color(green)(x=0)
[1]" "f(0)=(0)^2-1
[2]" "f(0)=0-1
[3]" "color(green)(f(0)=-1)
color(orange)(x=2)
[1]" "f(2)=(2)^2-1
[2]" "f(2)=4-1
[3]" "color(orange)(f(2)=3)
Now that you have solved for all the possible values of f(x), your range is:
R={-1,0,3}