학습 · 대수학 2

Rational Functions: Asymptotes, Holes, and the Forbidden Inputs

A rational function is a fraction of polynomials. Master vertical/horizontal asymptotes, where they come from, and the difference between an asymptote and a hole.

8 분 TEKS 6M,6N,6O 대수학 2

Functions with forbidden inputs

A rational function is one polynomial divided by another. Wherever the denominator is zero, the function is undefined — and either has a vertical asymptote or a hole. Reading these features off the equation is the entire game.

Vertical asymptotes

f(x) = 1 / (x − 2) Denominator zero at x = 2 → vertical asymptote x = 2 The function blows up to ±∞ as x approaches 2.
x = 2
Vertical asymptote where the denominator is zero. The graph never touches the dashed line.
Practice

Find the vertical asymptote

What is the vertical asymptote of f(x) = 1 / (x − 2)?

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Horizontal asymptotes

What happens as x goes to ±∞ depends on the degrees of numerator and denominator.

deg(num) < deg(den): y = 0 deg(num) = deg(den): y = ratio of leading coefficients deg(num) > deg(den): no horizontal asymptote (oblique or no asymptote)

Example

f(x) = (2x + 1) / (x − 4) Both degree 1; ratio of leading coefficients = 2/1 Horizontal asymptote: y = 2
Practice

Find the horizontal asymptote

For f(x) = (2x + 1) / (x − 4), what is the horizontal asymptote?

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Holes vs asymptotes

If a factor cancels between numerator and denominator, you get a hole, not an asymptote. The function has the same shape as the simplified form, except for one missing point.

f(x) = (x² − 9) / (x + 3) = (x − 3)(x + 3) / (x + 3)    (factor) = x − 3    (cancel, but x ≠ −3) Graph of x − 3 (a line) with a hole at x = −3.
Practice

Simplify and identify the hole

Simplify (x² − 9) / (x + 3).

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3-second recap

  • Vertical asymptote: denominator zero (and numerator non-zero) at that x.
  • Hole: factor cancels — same x, but no asymptote.
  • Horizontal asymptote: compare degrees; equal → leading-coefficient ratio.
  • Always check the original domain — canceled factors still exclude their zeros.