How do you find the domain and range without having to graph? Like, is their an algebraic method? For example if I am asked to find the domain of $g(t) = \sqrt{t^2 + 6t}$ , how do I determine the range of this? 
Is their a universal algebraic method that I don't know about? 

Writing $g(t)=\sqrt{t(t+6)}$, you can see that the domain must be restricted to values of $t$ for which $t(t+6)\geq 0$. You can solve this inequality to see that the domain then is $(-\infty,-6]\cup[0,\infty)$. This just excludes the open interval $(-6,0)$ on which $t(t+6)<0$.
So what is the image of $g$? (Note: the range can be taken to be $(-\infty,\infty)$, but I think you want only the more restricted set of values actually assumed by $g$; the terminology is not standardized, unfortunately--but I know what you mean.)  Certainly $g(0)=0$, and all values of $g$ are in $[0,\infty)$ because the square root function always returns a nonnegative value. But does $g$ assume all values in $[0,\infty)$? The answer is "yes". You will have to accept that $g(t)$ can be made as large as you like by taking $t$ sufficiently large, and there are no "holes" in the image because $g((-\infty,-6])$ meets $g([0,\infty))$.
