I'm working on some homework, and my google-fu isn't helping me out with this question. I've already solved it, but I need to put it into the proper notation in order to get full credit. I've already figured out that the range is $[-5,\infty)$, but I can't figure out the proper notation, nothing looks right to me. I feel like the answer should be $\mathbb R\setminus \{(-\infty, -5)\}$ ?
(the question in question: Find the largest possible subset $A$ of $\mathbb R$ that will make the following functions well-defined. $A\to \mathbb R$ given by $f(x) = \sqrt{3x+15}$: The range where the question is well defined is from $-5$ to infinity, but I don't know how to "formally" say it.
$$f(x) = \sqrt{3x+15}$$
This function is well-defined on $[-5,\infty)$, just as you wrote. That was already the correct final notation. You could also write $[-5,\infty) \subset \mathbb{R}$, or you could write $\mathbb{R}\setminus (-\infty,-5)$.
But $\{(-\infty,-5)\}$ is the set containing an interval, not the interval itself. You could write $\{x \in (-\infty,-5)\}$, but that is equivalent to $(-\infty,-5)$.
Finally, the word "range" when involving functions means the set of possible outputs of a function. From context clues, I gather that you mean "interval" when you used the word range. That is a more appropriate term in this context.