I graphed 3x^2 + 3x - 18 = 0 and am thinking that the answer might be E. "Either I or III" because it looks like (-infinity, -3] is monotonic decreasing and [2, infinity) is monotonic increasing. I'm not sure that this was the correct method, though, and it makes a funny looking function that looks like vertical lines!

Yes choice E) "Either I or III" is the correct answer. You solve 3x^2 + 3x - 18 = 0 to get the two solutions x = -3 or x = 2. Those will set up the boundary points to be used in a sign chart for the first derivative test

After doing the first derivative test, you'll find that for f(x)...
* The interval (-infinity, -3) is increasing
* The interval (-3,2) is decreasing
* The interval (2,infinity) is increasing

Which matches with what statements I and III are saying. Statement II is false because [-3, infinity) represents a mix of increasing and decreasing intervals on f(x). The graph fails the horizontal line test for this interval. For an interval like (-infinity, -3], the graph passes the horizontal line test.

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Note: recall that the horizontal line test is the idea that if you can pass a horizontal line through more than one point on the curve, then it fails the test. This means the graph is not one-to-one (injective). If it passes the horizontal line test, then the graph is one-to-one.