1.9 Rational Functions and Vertical Asymptotes

Rational Functions Defined

A rational function is a ratio of two polynomials, f(x) = p(x)/q(x), where q(x) ≠ 0. Its domain excludes all x-values that make the denominator zero. Analyzing where the denominator vanishes helps reveal key features like holes and vertical asymptotes, which describe the function's end behavior near those excluded x-values.

Vertical Asymptotes vs. Holes

If a factor (x−a) is in the denominator but not canceled by a matching factor in the numerator, x = a is a vertical asymptote — the function's output grows without bound as x approaches a. If the factor cancels with the numerator, there's a removable discontinuity (hole) at x = a instead, since the simplified function is defined there (just not the original).

Behavior Near a Vertical Asymptote

Near a vertical asymptote at x = a, f(x) approaches positive or negative infinity. The sign on each side depends on the multiplicity of the factor (x−a): odd multiplicity causes the graph to shoot to opposite infinities on either side, while even multiplicity sends it to the same infinity on both sides.

Key terms

Rational function

A function of the form f(x) = p(x)/q(x), where p and q are polynomials and q(x) ≠ 0.

Vertical asymptote

A vertical line x = a where the function's value grows unboundedly (±∞) because the denominator equals zero and isn't canceled by the numerator.

Removable discontinuity (hole)

A point where a factor cancels between numerator and denominator, leaving the simplified function defined but the original undefined at that x-value.

Effect of odd multiplicity at an asymptote

The function approaches opposite infinities (one side +∞, other side −∞) around the vertical asymptote.

Effect of even multiplicity at an asymptote

The function approaches the same infinity (+∞ or −∞) on both sides of the vertical asymptote.

Try an AP-style question

Where are the vertical asymptotes of 1/(x² − 9)?

1. y = 0
2. x = 9
3. x = 3 only
4. x = 3 and x = −3