The domain is the "allowed" values for t (the input of the function) and the range is the values that can be returned by the function given that you use the appropriate values for t.
Assuming you are working strictly with the real numbers, RR, then:
For the domain:
We can't have t=0 on the bottom otherwise we would be dividing by 0. Also we can't take the square root of a negative so anything less than or equal to 0 is not allowed. (i.e 0 is the lowest allowed number on the domain). There is no limit on the highest number so the domain is:
tin (0,oo)
For the range:
We have already deduced that the lowest value for t is 0. We find that as we make t very small the fraction becomes very big, in fact f(t) gets infinitely big as t approaches zero. (Don't forget the negative). So the lower limit on our range will be -oo. On the contrary, as we make t infinitely big, the larger value on our domain, f(t) gets very small and approaches 0 when t gets to infinity. So the other limit on our range is 0 Hence the range is:
f(t) in (-oo,0)
Note, we have used curved brackets to indicate an "open interval." This means the limit specified is NOT part of the set. That is: t can be greater than 0 but not equal to 0.
Indeed there do exist cases where the limiting numbers are also part of the domain or range, in which case we use a square bracket [x,y] to indicate a closed interval.
