Chapter 4: Q. 24 (page 234)

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$H (x) = \frac{x^{2} + 4}{x^{4} - 1}$

Short Answer

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The graph is

Step by step solution

Given information

Given function $H (x) = \frac{x^{2} + 4}{x^{4} - 1}$

Factoring the numerator and denominator

Factoring, we get

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

Calculate the domain

Setting denominator to zero, we get

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

The domain is all real numbers except $1, - 1$

Calculate the x intercept

Setting numerator to zero, we get

$x^{2} + 4 = 0 x^{2} = - 4$

There is no $x$ intercept

The $y$ intercept is $H (0) = - 4$

Calculate the vertical asymptote

Setting denominator to zero, we get

$x^{2} - 1 = 0 x^{2} = 1 x = 1, - 1$ is the vertical asymptote

Calculate the horizontal asymptote

The degree of numerator is less than denominator so the horizontal asymptote is $y = 0$

The intersection point is

$H (x) = \frac{x^{2} + 4}{x^{4} - 1} = 0 x^{2} + 4 = 0 x^{2} = - 4$

It doesn't intersects $H$

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

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculate the domain

Calculate the x intercept

Calculate the vertical asymptote

Calculate 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.H(x)=x2+4x4-1

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculate the domain

Calculate the x intercept

Calculate the vertical asymptote

Calculate the horizontal asymptote

Sketching the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Calculus

Discrete Mathematics

Mechanics Maths

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