Lesson 3.1

Domain Restrictions

A rational expression is a fraction where the numerator and denominator are polynomials. The very first rule: you can never divide by zero.

Introduction

Welcome to Unit 3: Rational Functions! Every rational expression has a domain — the set of all -values that produce a valid output. Any value that makes the denominator equal to zero is excluded from the domain.

Past Knowledge

Factoring polynomials (Unit 2) and solving equations.

Today's Goal

Find the domain of any rational expression by setting the denominator equal to zero and solving.

Future Success

Domain restrictions will reappear as vertical asymptotes, holes, and extraneous solutions throughout Unit 3.

Key Concepts

What Is a Rational Expression?

Just like a fraction with numbers — except the numerator and denominator contain variables.

Examples vs. Non-Examples

✓ Rational

✓ Rational

✗ Not Rational

✗ Not Rational

Finding Domain Restrictions

Set the denominator = 0 and solve

1

Factor the denominator completely

2

Set each factor equal to zero

3

Those solutions are excluded from the domain

Domain Notation

Two ways to state restrictions:

Set-builder:

Interval:

Worked Examples

Example 1: Linear Denominator

Basic

Find the domain of .

1

Set denominator = 0

Domain: all real numbers,

Example 2: Quadratic Denominator

Intermediate

Find the domain of .

1

Factor the denominator

2

Set each factor = 0

Domain: all real numbers,

Example 3: Higher-Degree Denominator

Advanced

Find the domain of .

1

Factor the denominator

2

Set each factor = 0

Domain: all real numbers,

Common Pitfalls

Only Looking at the Numerator

Restrictions come from the denominator only. The numerator can be zero — that just gives , which is perfectly fine.

Forgetting to Factor

has two solutions, not one. Always factor the denominator completely before finding restrictions.

Real-Life Applications

In electrical engineering, the impedance of a circuit is modeled with rational expressions. Division by zero corresponds to a resonance frequency — a point where the circuit theoretically produces infinite output. Knowing domain restrictions helps engineers identify and avoid dangerous resonance conditions.

Practice Quiz

Loading...