Domain And Range Of Rational Functions - Varsity Tutors

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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 xUse 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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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

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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.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

A$\text{All real numbers except } x = 3$B$\text{All real numbers except } x = 0$C$\text{All real numbers}$D$\text{All negative numbers}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You are designing a roller coaster, and the height function is $h(t) = \frac{100}{t-5}$. Find the domain.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$\text{All real numbers except } y = 0$B$\text{All real numbers}$C$\text{All positive numbers}$D$\text{All real numbers except } y = 2$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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