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Chapter 4: Q. 41 (page 234)

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.

f(x)=x2+1x

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

Given functionf(x)=x2+1x

02

Factoring the numerator and denominator

Factoring, we get

f(x)=x2+1xf(x)=x3+1xf(x)=(x+1)x2-x+1x

03

Calculate the domain

Setting denominator to zero, we get

x=0

The domain is all real numbers except0

04

Calculate the x intercept

Setting numerator to zero, we get

(x+1)x2-x+1=0x2-x+1=0x+1=0x=-1

There is no yintercept asf(0)=undefined

05

Calculate the vertical asymptote

The zeroes of denominator arex=0so the vertical asymptote isx=0

06

Calculating the horizontal asymptote

The degree of numerator is greater than denominator so there is an oblique asymptote

f(x)=x2+1x

y=x2is an oblique asymptote

f(x)=(x+1)x2-x+1x=x2(x+1)x2-x+1=x3x3-x2+x+x2-x+1=x30=-1

So it doesn't intersectf

07

Sketching the graph

The graph is

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