Functions f and g intersect at the points (–3,5), (2,8) and (11,–17). Which of the following could define the function h(x) = f(x) – g(x)?

A) h(x) = (x + 3)(x – 2)(x –11)
B) h(x) = (x – 5)(x – 8)(x + 17)
C) h(x) = (x + 5)(x – 6)(x + 6)
D) h(x) = (x – 3)(x + 2)(x + 11)

The idea here—and it’s a tricky one—is that when you add or subtract functions, you’re doing that to the y-values for each x-value. For example, let’s say we have functions a and b, and that function a contains points (1, 3) and (2, –1) while function b contains points (1, 1) and (2, 6). If we added a(x) and b(x), the resulting function would contain points (1, 4) and (2, 5)—we sum the y-values at each x-value. Hopefully the illustration using linear functions below isn’t too noisy to look at.

In the above, a(x)=-4x+7 and b(x)=5x-4 contain the points named above for functions a and b. The sum of those is a(x)+b(x)=-4x+7+5x-4=x+3, and it contains (1, 4) and (2, 5).

With that understanding we can solve this problem. When we know that two functions intersect at a point, then we know their y-values are equal at that point. Therefore, we know that if we subtract one intersecting function from the other, the resulting function should have a y-value of zero at that x-value. In other words, if functions f and g intersect at (–3, 5), (2, 8) and (11, –17), we should expect h(x)=f(x)-g(x) to contain the points (–3, 0), (2, 0), and (11, 0).

Choice A does that for us. It will have x-intercepts at –3, 2, and 11.

One final note about the answer choices, which of course are always a potential aid in a multiple choice test. Looking at the choices here, all are in factored form to highlight their x-intercepts, and 3 of them are built up from numbers in the intersection points we’re given. That alone won’t get you all the way there, but might get you thinking along the right lines. When you’re stumped, look to the answer choices for a little inspiration.

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