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Chapter 4: Q. 57 (page 236)

Graph each of the following functions:

y=x2-1x-1y=x3-1x-1y=x4-1x-1y=x5-1x-1

Is x=1a vertical asymptote? Why not? What is happening for x=1? What do you conjecture about y=xn-1x-1;n鈮�1an integer, for x=1?

Short Answer

Expert verified

The function is undefined at x=1, so there is no vertical asymptote there. There is a hole atx=1.

Step by step solution

01

Step 1. x=1 is not a vertical asymptote.

For all the function we are given here x=1is not a vertical asymptote. For vertical asymptote we put denominator equal to zero.

We see in every function it seems that vertical asymptote will be x=1but this is not so. Let us draw the graph of all these and then make conclusions. Also, as we write the function, we observe that in each has a common factor in numerator and denominator.

From this we come to the conclusion that x=1is not a vertical asymptote rather it is a hole.

02

Step 2. The graph of y=x2-1x-1 is

03

Step 3. The graph of y=x3-1x-1 is

04

Step 4. The graph of y=x4-1x-1 is

05

Step 5. The graph of y=x5-1x-1 is

06

Step 6. Explanation for x=1.

For x=1the function is undefined, so there is no vertical asymptote there.

Now for the function

y=xn-1x-1;n鈭�I鈭赌n鈮�1

Here also numerator and denominator have a common factor.

This means x=1is a hole.

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Most popular questions from this chapter

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

G(x)=x3+1x2+2x

Find the real zeros of f. Use the real zeros to factor f.

f(x)=x3-8x2+17x-6

United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000cubic inches. See the illustration.

Part (a): Express the surface area S of the box as a function of x.

Part (b): Using a graphing utility, graph the function found in part (a).

Part (c): What is the minimum amount of cardboard that can be used to construct the box?

Part (d): What are the dimensions of the box that minimize the surface are?

Part (e): Why might UPS be interested in designing a box that minimizes the surface area?

In Problems 63鈥�72, find the real solutions of each equation.

x4-2x3+10x2-18x+9=0

In Problems 57鈥� 62, find the real zeros of f. If necessary, round to two decimal places.

f(x)=x4+1.2x3-7.46x2-4.692x+15.2881
See all solutions

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