The question is how many times does the graph of the rational function g(x)=(6x^2 - 5x - 6) / (6x^2 -13x + 6) cross the x-axis?
The answer was given already to be that it crosses the x-axis exactly once but I cant find out why.
I've tried using b^2 -4ac on the numerator but that tells me it crosses twice. So I figured maybe there is an asymptote that prevents the second x-intercept from occurring.
Can somebody explain why this question is so weird?
My impression is that this is a tricky trick question (sort of a double-cross), possibly intended to teach a valuable lesson about caution.
By asking only how many times the graph crosses the axis, it gives the impression that you should be able to answer by a "trick" in the positive sense, without having to do all the work. You could even call it a temptation to do so. And that is what the OP did, using the discriminant to find the number of zeros
of the numerator without actually solving or even factoring. That sounds like a smart thing to do, but isn't -- it's too smart for one's own good.
It turns out that the trick fails because one of the two apparent crossings corresponds not to an x-intercept of the rational function, but to a "hole", where both the numerator and the denominator are zero. It's necessary to look at the whole thing in order to be sure what happens.
Now, the kind of trick question that really gets me is one where such a trick (bypassing an actual solution) is what they
want you to do, but it turns out that "their" answer is
wrong, because the solution turns out not to exist at all. This is similar to those, but since "their" answer is correct, it looks like they know that you have to graph the entire function, or at least think thoroughly about it rather than take the misleading shortcut. That's tricky, but not evil ...
But, yes, I'd like to see the context, to see how much enticement there was for falling into the trap.