Functions
AI Assistant

Rational Functions

Asymptotes, holes, and the behavior of fractions

A rational function is a fraction of two polynomials, like f(x) = 1/x. The graph shows the classic hyperbola shape — the curve gets closer and closer to the axes but never touches them. Those invisible boundary lines are called asymptotes.

Vertical asymptotes occur where the denominator equals zero (the function is undefined). Horizontal asymptotes describe the function's behavior as x gets very large. Sometimes a factor cancels, creating a hole instead of an asymptote.

Ask the AI "Where is the asymptote?" or "Graph (x+1)/(x−2) and find its domain."

Graph

FAQ

What is a vertical asymptote?
A vertical asymptote is a vertical line x = a where the function approaches ±∞. It occurs where the denominator equals zero (and the numerator doesn't). For 1/x, the vertical asymptote is x = 0.
What is a horizontal asymptote?
A horizontal asymptote is the value y approaches as x → ±∞. For 1/x, y → 0 as x gets very large. Compare the degrees of numerator and denominator: same degree → ratio of leading coefficients; lower numerator degree → y = 0.
What is a hole in a rational function?
A hole occurs when a factor cancels from both numerator and denominator. For example, (x−2)(x+1)/(x−2) simplifies to x+1, but there's a hole at x = 2 because the original function is undefined there.
What is the domain of a rational function?
The domain is all real numbers except where the denominator is zero. For 1/(x−3), the domain is all x ≠ 3. For 1/((x−1)(x+2)), the domain excludes x = 1 and x = −2.

Related Topics

Polynomial FunctionsThe HyperbolaWhat is a Limit?