16Hit1600
Advanced MathVery hardHole and asymptote of a rational function

Advanced Math practice question

The function r is defined by r(x) = (2x − 6)/(x² − 9). In the xy-plane, the graph of y = r(x) has a hole at x = a and a vertical asymptote at x = b. What is the value of a + b?

Answer: 0

r(x) = 2(x − 3)/((x − 3)(x + 3)). The common factor x − 3 makes a hole at a = 3; the remaining x + 3 makes an asymptote at b = −3. So a + b = 0.

Common wrong answers

If you answered 3
3 is a alone.
If you answered 6
6 treats both x = 3 and x = −3 as asymptotes with the wrong signs.
If you answered -6
−6 has the hole and the asymptote swapped and negated.

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