Determine the domain and range of the following relations using set builder notation. 

I think I am starting to get the hang of using set builder notation, but I'm getting confused with the end points of these two specific questions. The blue text is hat I have already, but it would be greatly appreciated if someone could attempt at clearing up my confusion with the arrow endpoints. Thanks!

The "arrow endpoints" just show that the curve continues in the same way forever, to $\infty$ or $-\infty$ as the case may be. In both graphs the arrow endpoint shows that the graph continues down, and in the second graph it also continues to the left.
In your first graph, the range is correct but the domain is wrong. You see from the graph that $x$ can be from $-3$ up to $4$, so
$$D=\{x\mid -3\le x\le 4, x\in\Bbb R\}$$
The statement $x\in\Bbb R$ is usually left out, understood when working with two-dimensional graphs.
Note that the range has no smallest value for $y$, just the largest value of $3$. This is because of the arrow endpoint, which shows that $y$ continues down to $-\infty$. Is that clear?
The second graph has the correct domain and range.
