Chapter 4: Q. 16 (page 234)

In Problems $7 - 44$ , follow Steps $1$ through 7 on page $228$ to analyze the graph of each function.
$G (x) = \frac{x^{3} + 1}{x^{2} + 2 x}$

Short Answer

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The graph is

Step by step solution

Given information

Given the function $G (x) = \frac{x^{3} + 1}{x^{2} + 2 x}$

Factor the numerator and denominator

Factoring , we get

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

Calculate the domain

Substitute , we get

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

The domain is all real numbers except $0, - 2$

Substitute numerator to zero to get x intercept

Substituting, we get

$x^{3} + 1 = 0 x^{3} = - 1 x = - 1$

The $x$ intercept is $(- 1, 0)$

Calculate the vertical asymptote

Setting denominator to zero, we get

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

Calculating the oblique asymptote

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

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

So $y = x - 2$ is the oblique asymptote

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.
$G (x) = \frac{x^{3} + 1}{x^{2} + 2 x}$

Short Answer

Step by step solution

Given information

Factor the numerator and denominator

Calculate the domain

Substitute numerator to zero to get x intercept

Calculate the vertical asymptote

Calculating the oblique asymptote

Sketching the graph

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In Problems 7-44, follow Steps 1through 7 on page 228to analyze the graph of each function.G(x)=x3+1x2+2x

Short Answer

Step by step solution

Given information

Factor the numerator and denominator

Calculate the domain

Substitute numerator to zero to get x intercept

Calculate the vertical asymptote

Calculating the oblique asymptote

Sketching the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Statistics

Calculus

Theoretical and Mathematical Physics

Mechanics Maths

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.
$G (x) = \frac{x^{3} + 1}{x^{2} + 2 x}$