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Domain and Range of Rational Functions

Learn how to determine the domain and range of rational functions.

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Domain and Range of Rational Functions

Study Guide

Key Definition

For a rational function f(x)=p(x)/q(x), the domain is all real numbers except zeros of q(x). The range is all real numbers except values y for which the equation y·q(x)−p(x)=0 has no real solution.

Important Notes

A rational function is undefined where its denominator equals zero.
Graphing can help visualize the domain and range.
Holes occur where factors cancel each other in the numerator and denominator.
Always check for vertical asymptotes.
The range can vary based on the function's behavior.

Mathematical Notation

$\frac{a}{b}$ represents a fraction

$x^2$ represents x squared

$\sqrt{x}$ represents the square root of x

$f(x)$ represents a function of x

Use proper notation when solving problems.

Why It Works

Finding q(x)=0 gives domain restrictions; solving y·q(x)−p(x)=0 and analyzing behavior gives range restrictions.

Remember

Domain excludes values where q(x)=0.

Quick Reference

Domain:All real numbers except where $q(x) = 0$

Range:All real numbers except values y with no solution to y·q(x)−p(x)=0

Understanding Domain and Range of Rational Functions

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Beginner

Start here! Easy to understand

Beginner Explanation

For f(x)=1/x, the domain excludes x=0 because division by zero is undefined. The range excludes y=0 since 1/x never equals zero. Graphically, there is a vertical asymptote at x=0 and a horizontal asymptote at y=0.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the domain of $f(x) = \frac{1}{x-3}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are designing a roller coaster, and the height function is $h(t) = \frac{100}{t-5}$. Find the domain.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the range of $f(x) = \frac{x+2}{x-1}$.

Challenge Quiz

Single Choice Quiz

Advanced

What is the range of $f(x) = \frac{2x}{x^2 - 4}$?

Recap

Watch & Learn

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