Rational Functions

Definition

A rational function is a function of the form $R(x) = \frac{p(x)}{q(x)}$ , where $p$ and $q$ are polynomial functions and $q$ is not the zero polynomial.

Why It Matters

Asymptotes are the “no-go zones” and “speed limits” of the mathematical and physical world. Understanding these properties is essential for predicting steady-states in economics and biology. If you miscalculate an asymptote, you are fundamentally misjudging the ultimate limit of a system’s growth or the point where its internal logic breaks down.

Core Concepts

Domain: All real numbers except the zeros of the denominator $q(x)$ .
How to read: “All real numbers except the zeros of Q of x.”
- Meaning: The function is undefined where the denominator is zero (potential vertical asymptotes or holes).
Vertical Asymptotes (VA): Lines where $q(c) = 0$ ; graph approaches $\pm \infty$ .
How to read: “The vertical asymptotes occur where Q of c equals zero; the graph approaches positive or negative infinity.”
- Meaning / when to use: As x approaches a root of the denominator, the function value grows without bound. Key for sketching and understanding limits at infinity or at poles.
Horizontal (HA) and Oblique Asymptotes (OA): Long-term behavior ( $x \to \pm \infty$ ) based on the degrees of $p$ and $q$ .
How to read: “The horizontal or oblique asymptotes describe long-term behavior as x approaches positive or negative infinity, determined by comparing degrees of P and Q.”
- Meaning / when to use: The end behavior “speed limit” of the rational function. If deg(P) < deg(Q) → HA at y=0; deg(P)=deg(Q) → HA at ratio of leading coeffs; deg(P)=deg(Q)+1 → oblique (slant) asymptote found by polynomial division.
Holes: Occur when a factor $(x-c)$ is common and cancels.
How to read: “The holes occur when a factor the quantity x minus c is common to numerator and denominator and cancels.”
- Meaning: A common factor that is removed creates a hole (removable discontinuity) at x=c rather than an asymptote. The simplified function is used for graphing except at the hole.

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes