The domain is the set of xx values a function can take to give a real yy value, which in the function y = x^2 -5y=x2−5 is simply any xx value. For instance, when x=-6x=−6 then y = 36-5 = 31y=36−5=31. Similarly, when x=1000x=1000, then y=1000000-5=999995y=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.
