Chapter 4: Q. 43 (page 234)

In Problems $7 - 44$ , follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$f (x) = x + \frac{1}{x^{3}}$

Short Answer

Expert verified

The graph is

Step by step solution

Given information

Given function $f (x) = x + \frac{1}{x^{3}}$

Factoring the numerator and denominator

Factoring, we get

$f (x) = x + \frac{1}{x^{3}} f (x) = \frac{x^{4} + 1}{x^{3}}$

Calculate the domain

Setting the denominator to zero, we get

$x^{3} = 0 x = 0$

The domain is all real numbers except $0$

Calculate the x intercept

Setting numerator to zero, we get

$x^{4} + 1 = 0 x^{4} = - 1$

There is no $x, y$ intercept

Calculate the vertical asymptote

The zeroes of denominator is $x = 0$ so the vertical asymptote is $x = 0$

Calculating the horizontal asymptote

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

$f (x) = x + \frac{1}{x^{3}}$

The oblique asymptote is $y = x$

$f (x) = \frac{(x^{4} + 1)}{x} = x x^{4} + 1 = x^{2} x^{4} - x^{2} = - 1$

It doesn't intersects $f$

Sketching the graph

The graph is

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems $7 - 44$ , follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$f (x) = x + \frac{1}{x^{3}}$

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculate the domain

Calculate the x intercept

Calculate the vertical asymptote

Calculating the horizontal asymptote

Sketching the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Discrete Mathematics

Geometry

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.f(x)=x+1x3

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculate the domain

Calculate the x intercept

Calculate the vertical asymptote

Calculating the horizontal asymptote

Sketching the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Discrete Mathematics

Geometry

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

In Problems $7 - 44$ , follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$f (x) = x + \frac{1}{x^{3}}$