The domain is the set of #x# values a function can take to give a real #y# value, which in the function #y = x^2 -5# is simply any #x# value. For instance, when #x=-6# then #y = 36-5 = 31#. Similarly, when #x=1000#, then #y=1000000-5=999995#.
Therefore, #-oo < x < oo, x in RR#.
However, for #x in RR#, #x^2 >= 0#. In other words, a square number is always positive (greater than 0), so a square number minus five must be always greater than minus five. So,
#x^2 >= 0#
#:.#
#x^2 - 5 >= -5#
#:.#
#y >= -5#
This is the range of the function, which is defined as the set of #y# values that can be taken by the function. You'll never find a (real) solution for anything less than #y = -5#, for which #x = 0#.
