Properties of Rational Functions (Precalculus, Section 4.5)

Polynomial and Rational Functions

Introduction

This section explores the properties of rational functions, a key topic in precalculus. Rational functions are defined as quotients of polynomial functions, and their behavior is characterized by their domains, asymptotes, and graphical features.

Properties of Rational Functions

Definition of Rational Function

• Rational Function: A rational function is any function that can be written in the form: $R(x) = \frac{p(x)}{q(x)}$ where p(x) and q(x) are polynomial functions and q(x) is not the zero polynomial.

• Domain: The domain of a rational function is the set of all real numbers except those for which the denominator q(x) is zero.

Finding the Domain of a Rational Function

The domain of a rational function excludes any real number that makes the denominator zero.

• Example 1: $R(x) = \frac{3x^2 - 5}{x + 2}$ The denominator is zero when $x = -2$. Domain: $\{x \mid x \neq -2\}$

• Example 2: $R(x) = \frac{1}{x^2 - 25} = \frac{1}{(x - 5)(x + 5)}$ The denominator is zero when $x = 5$ or $x = -5$. Domain: $\{x \mid x \neq 5, x \neq -5\}$

• Example 3: $R(x) = \frac{2x^3}{x^2 + 1}$ The denominator $x^2 + 1$ is never zero for real $x$. Domain: All real numbers.

• Example 4: $R(x) = \frac{x^2 - 4}{x - 2}$ The denominator is zero when $x = 2$. Domain: $\{x \mid x \neq 2\}$

Additional info: Even if the numerator and denominator share a factor, the domain is determined by the denominator before simplification.

Graphing Rational Functions

Graphing rational functions involves understanding their intercepts, symmetry, and behavior near undefined points.

• Example: $H(x) = \frac{1}{x^2}$

  • Domain: All real numbers except $x = 0$.

  • Intercepts: No y-intercept (since $x = 0$ is not in the domain); no x-intercept (since $H(x) = 0$ has no real solution).

  • Symmetry: $H(-x) = H(x)$, so the function is even and its graph is symmetric about the y-axis.

Behavior Near Undefined Points

• As $x$ approaches 0 from either side, $H(x)$ becomes very large (unbounded) in the positive direction. This is written as $\lim_{x \to 0} H(x) = \infty$.

• As $x$ becomes very large (positive or negative), $H(x)$ approaches 0. This is written as $\lim_{x \to \infty} H(x) = 0$ and $\lim_{x \to -\infty} H(x) = 0$.

Summary Table: Domains of Example Rational Functions

Function	Domain
$R(x) = \frac{3x^2 - 5}{x + 2}$	$x \neq -2$
$R(x) = \frac{1}{x^2 - 25}$	$x \neq 5, x \neq -5$
$R(x) = \frac{2x^3}{x^2 + 1}$	All real numbers
$R(x) = \frac{x^2 - 4}{x - 2}$	$x \neq 2$

Key Properties of Rational Functions

• Vertical Asymptotes: Occur at values of $x$ where the denominator is zero and the numerator is not zero.

• Horizontal Asymptotes: Determined by the degrees of the numerator and denominator polynomials.

• Oblique (Slant) Asymptotes: Occur when the degree of the numerator is exactly one greater than the degree of the denominator.

• Symmetry: Even rational functions are symmetric about the y-axis; odd rational functions are symmetric about the origin.

Applications

• Rational functions model rates, ratios, and other phenomena in science and engineering.

• Understanding domains and asymptotes is essential for graphing and interpreting rational functions.