Rational Functions: Definitions, Domains, and Simplification

Rational Functions and Their Properties

Definition of Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is:

• Numerator: A polynomial $p(x)$

• Denominator: A polynomial $q(x)$, where $q(x) \neq 0$

The function is written as:

$f(x) = \frac{p(x)}{q(x)}$

Important: The denominator of a fraction cannot be zero.

Rational Equations vs. Rational Functions

• Rational Equation: An equation involving rational expressions, such as $12 = \frac{1}{x^2 - 1}$

• Rational Function: A function defined by a rational expression, such as $f(x) = \frac{1}{x^2 - 1}$

Restriction: Values of $x$ that make the denominator zero are excluded from the domain.

Domain: The set of all real numbers except those that make the denominator zero.

Finding the Domain of a Rational Function

To determine the domain of a rational function:

1. Set the denominator equal to zero and solve for $x$.

2. The domain is all real numbers except the solutions found in step 1.

Example:

Given $f(x) = \frac{1}{x^2 - 1}$, set $x^2 - 1 = 0$:

$x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1$

Domain: $\{x \mid x \neq 1, x \neq -1\}$

Writing Rational Functions in Lowest Terms

To simplify a rational function:

1. Factor the numerator and denominator.

2. Cancel any common factors.

3. Write the function in its simplest form.

Note: Always find the domain before simplifying the function.

Examples and Practice Problems

Function	Domain	Lowest Terms
$f(x) = \frac{3}{x^2 + 12}$	$\{x \mid x \neq \sqrt{-12}\}$ Additional info: Since $x^2 + 12 = 0$ has no real solutions, the domain is all real numbers.	$\frac{3}{x^2 + 12}$
$f(x) = \frac{x^2 + 9}{x - 3}$	$\{x \mid x \neq 3\}$	$\frac{x^2 + 9}{x - 3}$
$f(x) = \frac{6x}{2x^2 - 8}$	$\{x \mid x \neq 2, x \neq -2\}$	$\frac{3}{x - 2}$ Additional info: $2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)$, so after canceling $x + 2$ if present, the lowest terms is $\frac{3}{x - 2}$.
$f(x) = \frac{x^2 - 25}{x^2 - 25}$	$\{x \mid x \neq 5, x \neq -5\}$	$1$ Additional info: $x^2 - 25 = (x - 5)(x + 5)$, so numerator and denominator cancel, but domain excludes $x = 5$ and $x = -5$.

Summary Table: Steps for Rational Functions

Step	Description
1. Find Domain	Set denominator equal to zero and solve for $x$; exclude these values from the domain.
2. Simplify	Factor numerator and denominator, cancel common factors, and write in lowest terms.
3. State Final Function	Write the simplified function and its domain.

Key Terms

• Rational Function: A function of the form $f(x) = \frac{p(x)}{q(x)}$

• Domain: All real numbers except those that make the denominator zero

• Lowest Terms: The simplest form of a rational function after canceling common factors