Which of the following statements about a rational function is correct when a common…
MATHEMATICS · Single correct · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions
Problem
Which of the following statements about a rational function f(x)=Q(x)P(x) is correct when a common factor (x−a) cancels completely between P(x) and Q(x)?
(a) x=a lies in the domain after cancellation; the graph is smooth and continuous there.
(b) Cancelling (x−a) changes the factor into a vertical asymptote at x=a. (The graph shoots off to ±∞.)
(c) Although (x−a) cancels, x=a is still excluded from the domain, so the graph has a removable hole at that point.
(d) The cancellation affects neither the domain nor the range; the simplified expression describes the graph everywhere including x=a. (No special behaviour.)