Rational functions, asymptotes, and discontinuities

The parent rational function f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0: its graph gets closer and closer to those lines without touching them. The transformed function f(x) = a/(x - h) + k has a vertical asymptote at x = h and a horizontal asymptote at y = k. For a general rational function p(x)/q(x), first cancel any common factors: a canceled factor leaves a hole, a removable discontinuity, while a remaining zero of the denominator gives a vertical asymptote, an infinite discontinuity. For example, g(x) = (x^2 - 4)/(x - 2) simplifies to x + 2 with a hole at x = 2, while f(x) = (2x + 1)/(x - 3) has a vertical asymptote at x = 3. For the horizontal asymptote, compare degrees: if the numerator's degree is smaller, it is y = 0; if the degrees are equal, it is the ratio of the leading coefficients, so f(x) = (2x + 1)/(x - 3) has y = 2; and if the numerator's degree is exactly one larger, there is a slant asymptote instead. The domain excludes every value that makes the original denominator zero, and it can be written in interval notation, (-infinity, 3) U (3, infinity), in inequality notation, x < 3 or x > 3, or in set notation, {x | x is not 3}. Near a vertical asymptote, check each side separately: for f(x) = 1/(x - 3), as x approaches 3 from the right the outputs grow toward positive infinity, and from the left they fall toward negative infinity.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report