Chapter 4: Q. 13 (page 234)

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

Short Answer

Expert verified

The graph for the function $P (x) = \frac{x^{4} + x^{2} + 1}{x^{2} - 1}$ is,

Step by step solution

Step 1 Given function is,

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

which may also written as,

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

The domain of $P$ is, $\{x | x \neq \pm 1\}$ .

As $0$ is in the domain, substitute $x = 0$ in the given equation.

$P (0) = - \frac{1}{1} = - 1$

So, the obtained intercepts are, $(0, - 1)$ .

Step 2 The real zeros of the denominator are the real solutions of the equation.

$(x + 1) (x - 1) = 0 x = \pm 1$

The two lines $x = 1, x = - 1$ are the two vertical asymptotes of the graph.

In the given function, the degree of the numerator is greater than the degree of the denominator. So, $P$ is improper.

Let us use the long division method to find the horizontal asymptote.

Therefore, $P (x) = x^{2} + 2 + \frac{3}{(x + 1) (x - 1)}$

The obtained quotient is $x^{2} + 2$ . So, the graph does not have any horizontal or oblique asymptote.

The graph of $P$ will approach the graph of $y = x^{2} + 2$ as $x \to \infty$ .

Step 3 Substitute Px=x2+2 to determine whether the graph P intersects the horizontal asymptote.

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

There exists no solution. So, the graph of $P$ do not intersect $y = x^{2} + 2$ .

Step 4 Now plot a graph by using the known values.

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

Short Answer

Step by step solution

Step 1 Given function is,

Step 2 The real zeros of the denominator are the real solutions of the equation.

Step 3 Substitute Px=x2+2 to determine whether the graph P intersects the horizontal asymptote.

Step 4 Now plot a graph by using the known values.

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In Problems 7-44, follow Steps 1through 7on page 228to analyze the graph of each function.Px=x4+x2+1x2-1

Short Answer

Step by step solution

Step 1 Given function is,

Step 2 The real zeros of the denominator are the real solutions of the equation.

Step 3 Substitute Px=x2+2 to determine whether the graph P intersects the horizontal asymptote.

Step 4 Now plot a graph by using the known values.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

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