Rational Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rational Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A rational function is a ratio of two polynomials: f(x) = P(x)/Q(x) where P and Q are polynomials and Q(x) \neq 0.

Rational functions are the "fractions" of the function world — they behave like polynomials except near the zeros of the denominator, where they blow up or have holes.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Key behaviors are at zeros of Q (vertical asymptotes or holes) and as x \to \pm\infty (horizontal or oblique asymptotes, determined by degrees of P and Q).

Common stuck point: Holes vs. asymptotes: if a factor cancels, it's a hole, not an asymptote.

Sense of Study hint: Factor both numerator and denominator completely. Cancel common factors (those give holes), then set the remaining denominator to zero for asymptotes.

Worked Examples

Example 1

medium
Find the vertical and horizontal asymptotes of f(x) = \frac{3x + 2}{x - 4}.

Solution

  1. 1
    Vertical asymptote: set denominator = 0: x - 4 = 0, so x = 4.
  2. 2
    Horizontal asymptote: numerator and denominator have the same degree (both degree 1).
  3. 3
    The horizontal asymptote is y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{3}{1} = 3.

Answer

\text{Vertical: } x = 4, \quad \text{Horizontal: } y = 3
For rational functions, vertical asymptotes occur where the denominator is zero (and the numerator is not). The horizontal asymptote depends on comparing the degrees of numerator and denominator.

Example 2

hard
Find all asymptotes and holes of f(x) = \frac{x^2 - 9}{x^2 - x - 6}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the x-intercepts and vertical asymptotes of g(x) = \frac{x^2 - 1}{x^2 + 3x + 2}.

Example 2

hard
Find all asymptotes (vertical, horizontal, and oblique) of f(x) = \frac{x^2 + 2x - 3}{x - 1}. Does the function have a hole or a vertical asymptote at x = 1?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

polynomial functionsfractions