Introduction to Rational Functions: Definitions, Domains, and Simplification

Rational Functions

Definition and Structure

A rational function is a function that can be expressed as the quotient of two polynomials, where the numerator and denominator are both polynomials. The general form is:

• Numerator: Polynomial

• Denominator: Polynomial (cannot be zero)

Key Point: The denominator of a rational function must never be zero, as division by zero is undefined.

Example:

• Rational Equation: $\displaystyle 12 = \frac{1}{x-1}$

• Rational Function: $f(x) = \frac{1}{x-1}$

Restriction: $x \neq 1$

Domain: $x \in \mathbb{R}, x \neq 1$

Domain of Rational Functions

The domain of a rational function consists of all real numbers except those that make the denominator zero.

• To determine the domain, set the denominator equal to zero and solve for $x$.

• Exclude any $x$ values that make the denominator zero from the domain.

Example: For $f(x) = \frac{1}{x-1}$, the denominator $x-1 = 0$ when $x = 1$, so the domain is all real numbers except $x = 1$.

Simplifying Rational Functions

To write a rational function in lowest terms:

• Factor the numerator and denominator completely.

• Cancel any common factors.

• Always state the domain after simplification.

Example:

• $f(x) = \frac{x^2-1}{x-1}$

• Factor numerator: $x^2-1 = (x-1)(x+1)$

• Simplify: $\frac{(x-1)(x+1)}{x-1} = x+1$, for $x \neq 1$

Worked Examples

Example 1

• Given $f(x) = \frac{12}{x-1}$

• Domain: $x \neq 1$

• Lowest terms: Already simplified

Example 2

• Given $f(x) = \frac{x^2-1}{x-1}$

• Factor numerator: $x^2-1 = (x-1)(x+1)$

• Simplify: $\frac{(x-1)(x+1)}{x-1} = x+1$, for $x \neq 1$

Practice Problems

• Find the domain and write in lowest terms:

• $f(x) = \frac{12}{x-1}$

• $f(x) = \frac{x^2-1}{x-1}$

• $f(x) = \frac{x^2}{2x^2+2x-8}$

• $f(x) = \frac{2x-8}{2x^2+2x-8}$

Steps:

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

2. Exclude those $x$ values from the domain.

3. Factor numerator and denominator, then simplify.

Summary Table: Rational Functions

Function	Domain	Lowest Terms
$\frac{12}{x-1}$	$x \neq 1$	$\frac{12}{x-1}$
$\frac{x^2-1}{x-1}$	$x \neq 1$	$x+1$
$\frac{x^2}{2x^2+2x-8}$	$2x^2+2x-8 \neq 0$	Factor and simplify as needed
$\frac{2x-8}{2x^2+2x-8}$	$2x^2+2x-8 \neq 0$	Factor and simplify as needed

Additional info: The notes focus on the definition, domain, and simplification of rational functions, which are central topics in Precalculus Chapter 4 (Polynomial Functions and Rational Functions).