1.10 Rational Functions and Holes

Holes vs. Asymptotes

A rational function is a ratio of two polynomials. If a factor (x−a) cancels from both the numerator and denominator, the graph has a removable discontinuity (hole) at x = a, not a vertical asymptote. A vertical asymptote occurs where a factor remains uncanceled in the denominator after simplifying.

Finding the Hole's Location

To find a hole: factor numerator and denominator, cancel common factors, then plug the canceled zero's x-value into the simplified function to get its y-coordinate. The hole is the point (a, value) that's missing from the graph — the function is undefined there even though the simplified graph looks continuous.

Why Holes Occur

Holes arise because the original function is undefined wherever the denominator equals zero, even if algebraic cancellation removes that factor. The simplified expression is only equivalent to the original everywhere except at that x-value, so graphs of the original function must show an open circle there.

Key terms

Hole (removable discontinuity)

A point where a rational function is undefined because a factor canceled from numerator and denominator; graphed as an open circle.

How to find a hole's x-value

Set the canceled factor equal to zero and solve for x.

How to find a hole's y-value

Substitute the hole's x-value into the simplified (post-cancellation) function.

Hole vs. vertical asymptote

Hole: factor cancels from both numerator and denominator. Vertical asymptote: factor remains only in denominator after simplifying.

Why holes exist despite cancellation

The original function is still undefined at that x-value; canceling factors only simplifies the expression, it doesn't change the domain.

Try an AP-style question

Where is the hole in the graph of (x² − 1)/(x − 1)?

1. (1, 1)
2. (−1, 0)
3. (1, 0)
4. (1, 2)