Chapter 4.6, Problem 12E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Analyzing the Graph of a Function In Exercises 5-34, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = x 2 + 1 x 2 − 4

To determine

To graph: The function y=x2+1x24

Explanation

Given: The function y=x2+1x2âˆ’4.

Graph:

Consider the provided function.

The function being a rational function has a domain as all the real numbers except âˆ’2 and 2.

The highest negative value that the function can take is âˆ’0.25 and the function can also not take a positive value less than or equal to 1. This implies that the range of the function is (âˆ’âˆžâˆ’0.25]âˆª(1,âˆž).

Now find the x and y intercepts by equating f(x) and x to zero respectively to obtain:

There are no x-intercepts and the y-intercept is (0,âˆ’0.25).

The function has two vertical asymptotes x=2,x=âˆ’2 as the denominator is not defined at those points.

Also,

limxâ†’âˆž(x2+1x2âˆ’4)=limxâ†’âˆž(1+1x21âˆ’4x2)=1+limxâ†’âˆž1x21âˆ’limxâ†’âˆž4x2=1+01âˆ’0=1

This implies that the function has a horizontal asymptote as y=1.

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

âˆ’10x(x2âˆ’4)2=0x=0

This gives four test intervals (âˆ’âˆž,âˆ’2),(âˆ’2,0),(0,2),(2,âˆž).

Let âˆ’3âˆˆ(âˆ’âˆž,âˆ’2)

f'(âˆ’3)=âˆ’10(âˆ’3)((âˆ’3)2âˆ’4)2>0

The function is increasing in this interval.

Let âˆ’1âˆˆ(âˆ’2,0)

f'(âˆ’1)=âˆ’10(âˆ’1)((âˆ’1)2âˆ’4)2=109>0

The function is increasing in this interval.

Let 1âˆˆ(0,2)

f'(âˆ’1)=âˆ’10(1)((1)2âˆ’4)2=âˆ’109<0

The function is decreasing in this interval

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started