Rational functions

Functions with fractions

A rational function is a ratio of polynomials:

f(x) = p(x)/q(x), where q(x) ≠ 0

The simplest is the reciprocal function f(x) = 1/x.

The graph of 1/x

• As x → 0, the graph shoots up or down: a vertical asymptote at x = 0.
• As x gets very large, 1/x → 0: a horizontal asymptote at y = 0.
• The graph has two separate branches (a hyperbola).

Domain

The domain excludes zeros of the denominator.

f(x) = 3/(x − 2) → domain: x ≠ 2.

Vertical asymptotes and holes

Factor the numerator and denominator.

• If a factor cancels → a hole at that x.
• If a factor stays in the denominator → a vertical asymptote.

Example: f(x) = (x − 1)/[(x − 1)(x + 3)]

• (x − 1) cancels → hole at x = 1 (at y = 1/4)
• x = −3 remains → vertical asymptote

Horizontal asymptotes

Compare degrees of numerator (n) and denominator (m):

Example: f(x) = (2x + 1)/(x − 3) → n = m = 1 → y = 2; vertical asymptote x = 3.

Transformations of 1/x

f(x) = a/(x − h) + k

• Vertical asymptote: x = h
• Horizontal asymptote: y = k
• a stretches and/or reflects

Example: f(x) = 2/(x + 1) − 3 → asymptotes x = −1 and y = −3.

Intercepts

• x-intercepts: numerator = 0 (and not cancelled).
• y-intercept: f(0), if defined.

f(x) = (x − 4)/(x + 2): x-intercept 4; y-intercept −2.

Applications

• Average cost: C(x) = (500 + 3x)/x. As x grows, the average cost approaches $3 (the horizontal asymptote).
• Inverse variation: time to finish a job vs number of workers.
• Concentration when mixing solutions.

Worked examples

Example 1: Asymptotes of f(x) = 5/(x − 4)? x = 4, y = 0.

Example 2: Asymptotes of f(x) = (3x)/(x + 2)? x = −2, y = 3.

Example 3: Is there a hole in (x² − 9)/(x − 3)? Yes, at x = 3 (the graph is y = x + 3 with a hole at (3, 6)).

Common mistakes

• Calling every zero of the denominator an asymptote (it might be a hole).
• Wrong horizontal asymptote rule.
• Thinking graphs can never cross a horizontal asymptote (they sometimes can in the middle; asymptotes describe end behaviour).

Try it yourself

1. Domain of f(x) = 1/(x + 5)?
2. Asymptotes of f(x) = (4x − 1)/(2x + 6)?
3. Asymptotes of f(x) = −1/(x − 2) + 4?
4. Find any holes in f(x) = (x + 2)/(x² − 4).

Answers: 1) x ≠ −5 2) x = −3, y = 2 3) x = 2, y = 4 4) hole at x = −2 (vertical asymptote at x = 2)

Quick recap

• A rational function is a ratio of polynomials.
• Vertical asymptotes come from denominator zeros; cancelled factors give holes.
• Compare degrees to find the horizontal asymptote.