Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 73
Chapter 4, Problem 73

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)

Verified step by step guidance
1
Identify the domain of the function by finding the values of x that make the denominator zero. Solve x^2 - 4 = 0 to find these values.
Simplify the function if possible by factoring the numerator and denominator. Factor the denominator as (x-2)(x+2) and check for common factors with the numerator (x-2).
Determine the vertical asymptotes by setting the denominator equal to zero and excluding any values canceled out during simplification. These are the values where the function is undefined and the graph may have vertical lines.
Find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. Use the rules for rational functions to determine the asymptote equation.
Calculate key points such as intercepts by setting f(x) = 0 to find x-intercepts and evaluating f(0) to find the y-intercept. Plot these points along with the asymptotes to guide the sketch of the graph.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
15m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential because values that make the denominator zero are excluded and often correspond to vertical asymptotes or holes.
Recommended video:
6:04
Intro to Rational Functions

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.
Recommended video:
6:24
Introduction to Asymptotes

Graphing Steps for Rational Functions

Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes. Plotting key points and analyzing limits help sketch an accurate graph, following a systematic approach such as the seven-step method.
Recommended video:
8:19
How to Graph Rational Functions
Related Practice
Textbook Question
Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.
2
views
Textbook Question

In Exercises 73–74, use the graph of the rational function to solve each inequality.

1/4(x + 2) > - 3/4(x - 2)

Textbook Question

In Exercises 71–72, use the graph of the polynomial function to solve each inequality.


2x3 + 11x2 < 7x + 6

5
views
Textbook Question

In Exercises 73–74, use the graph of the rational function to solve each inequality.


1/4(x + 2) ≤ - 3/4(x - 2)

Textbook Question

In Exercises 71–72, use the graph of the polynomial function to solve each inequality.


2x3+11x2≥7x+62x^3 + 11x^2 ≥ 7x + 6

2
views
Textbook Question

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x2+x−6)