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? 

$t^2 + 6t \geq 0$. Thus $g(t) \geq 0$. This gives the range $[0, +\infty)$. For more "details" about the range, take a non-negative real number $r \geq 0$, then show that:
you can find an $t$ with $t \leq -6$ or $t \geq 0$ such that: $g(t) = r$. This translates to the equation:
$\sqrt{t^2 + 6t} = r \Rightarrow t^2 + 6t = r^2 \Rightarrow (t + 3)^2 - 9 = r^2 \Rightarrow (t + 3)^2 = r^2 + 9 \Rightarrow t + 3 = \pm \sqrt{r^2 + 9} \Rightarrow t = - 3 \pm \sqrt{r^2 + 9}$. Either value of $t$ just found works !
