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

Short Answer

Expert verified

The graph is

Step by step solution

Given information

Given function $f (x) = x^{2} + \frac{1}{x}$

Factoring the numerator and denominator

Factoring, we get

$f (x) = x^{2} + \frac{1}{x} f (x) = \frac{x^{3} + 1}{x} f (x) = \frac{(x + 1) (x^{2} - x + 1)}{x}$

Calculate the domain

Setting denominator to zero, we get

$x = 0$

The domain is all real numbers except $0$

Calculate the x intercept

Setting numerator to zero, we get

$(x + 1) (x^{2} - x + 1) = 0 x^{2} - x + 1 = 0 x + 1 = 0 x = - 1$

There is no $y$ intercept as $f (0) = u n d e f i n e d$

Calculate the vertical asymptote

The zeroes of denominator are $x = 0$ 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

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

$y = x^{2}$ is an oblique asymptote

$f (x) = \frac{(x + 1) (x^{2} - x + 1)}{x} = x^{2} (x + 1) (x^{2} - x + 1) = x^{3} x^{3} - x^{2} + x + x^{2} - x + 1 = x^{3} 0 = - 1$

So it doesn't intersect $f$

Sketching the graph

The graph is

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In Problems $7 - 44$ , follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$f (x) = x^{2} + \frac{1}{x}$

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

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Most popular questions from this chapter

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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

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

Geometry

Mechanics Maths

Applied Mathematics

Calculus

Discrete Mathematics

Logic and Functions

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