/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 92 Graph each rational function by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 92

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{-2 x^{2}}{x^{4}+1}$$

Short Answer

Expert verified
Domain: \((-\infty, \infty)\); Range: \((-\infty, 0]\); Horizontal asymptote at \(y=0\); Symmetric about y-axis.

Step by step solution

01

Domain Determination

To find the domain of the function \(f(x) = \frac{-2x^2}{x^4 + 1}\), identify values of \(x\) that do not make the denominator zero. Since \(x^4 + 1\) is always greater than zero for all real \(x\), the domain is all real numbers, or \((-\infty, \infty)\).
02

Range Determination

As the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the graph will approach the x-axis but never touch or cross it, implying that the range will be below the x-axis. The range of \(f(x)\) is \((-\infty, 0]\).
03

Vertical Asymptote Analysis

Vertical asymptotes occur where the denominator is zero. Since \(x^4 + 1\) is never zero, there are no vertical asymptotes for this function.
04

Horizontal Asymptote Analysis

The horizontal asymptote of a rational function is found by comparing the degrees of the polynomials. Since the degree of the numerator \(-2x^2\) is less than the degree of the denominator \(x^4 + 1\), the horizontal asymptote is at \(y = 0\).
05

Symmetry Analysis

Check for symmetry by testing if \(f(-x) = f(x)\) (even symmetry) or \(f(-x) = -f(x)\) (odd symmetry). Calculate \(f(-x) = \frac{-2(-x)^2}{(-x)^4 + 1} = \frac{-2x^2}{x^4 + 1} = f(x)\). Thus, the function is even and symmetric about the y-axis.
06

Graphing the Function

Begin by marking the x-axis as the horizontal asymptote. Plot several points across different values of \(x\) to observe the function's behavior (e.g., \(f(0) = 0\), \(f(1) = \frac{-2}{2} = -1\), \(f(-1) = \frac{-2}{2} = -1\)). Since the function is even, the left side of the graph mirrors the right side.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
When exploring rational functions, a crucial step is determining the domain and range. For the function given, \(f(x) = \frac{-2x^2}{x^4 + 1}\), the domain comprises all real numbers, expressed as \((-fty, fty)\). This is because the denominator, \(x^4 + 1\), remains positive for all \(x\) values, meaning it can never equal zero.
Understanding the range requires examining behavior as the x-values change. Here, the numerator's degree is less than the denominator's, resulting in the graph nearing but not crossing the x-axis. As such, the range is all real values below and including zero: \((-fty, 0]\). This indicates that the function's values are never positive.
Asymptotes
Asymptotes serve as invisible boundaries that guide a function's graph. With the given function, identifying vertical asymptotes involves locating points where the denominator equals zero. Since \(x^4 + 1\) is always positive, no vertical asymptotes exist for this function.
Horizontal asymptotes occur when examining the degrees of the numerator and denominator. In \(f(x) = \frac{-2x^2}{x^4 + 1}\), the numerator's polynomial degree of 2 is less than the denominator's degree of 4. This suggests the graph approaches the line \(y = 0\) as \(x\) moves towards infinity in either direction.
  • No vertical asymptotes
  • Horizontal asymptote at \(y = 0\)
Graph Symmetry
Graph symmetry can reveal much about the structure of a function. It can be determined by testing if the function replicates across either axis. For even symmetry, we test if \(f(-x) = f(x)\). In the equation \(f(x) = \frac{-2x^2}{x^4 + 1}\), substituting \(-x\) into the function yields the same result: \(\frac{-2x^2}{x^4 + 1}\).
This shows the function displays even symmetry, meaning the graph is symmetric about the y-axis. This type of symmetry simplifies graphing, as only half of the function needs to be plotted, with the other half mirroring it across the y-axis.
  • Even symmetry across the y-axis
Polynomial Degree Comparison
The degrees of the polynomials in a rational function impact its end behavior and asymptotes. For the function \(f(x) = \frac{-2x^2}{x^4 + 1}\), the numerator is of degree 2, while the denominator is degree 4.
When the numerator's degree is smaller, the horizontal asymptote tends towards \(y = 0\), as the value of the function diminishes towards zero with increasing \(|x|\). This comparison is essential for predicting how the function behaves when extending out towards positive or negative infinity.
  • Degree of numerator: 2
  • Degree of denominator: 4
  • Results in a horizontal asymptote at \(y = 0\)

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each equation and inequality. (These types of equations and inequalities occur in calculus.) (a) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}=0\) (b) \(\frac{\left(x^{2}-1\right)(1)-(x+1)(2 x)}{\left(x^{2}-1\right)^{2}}>0\)

Solve each rational inequality by hand. $$\frac{1}{x-3} \leq \frac{5}{x-3}$$

Train Curves When curves are designed for trains, sometimes the outer rail is elevated or banked so that a locomotive can safely negotiate the curve at a higher speed. Suppose a circular curve is being designed for a speed of 60 mph. The rational function \(f(x)=\frac{2540}{x}\) computes the elevation \(y\) in inches of the outer track for a curve with a radius of \(x\) feet, where \(y=f(x) .\) (Image can't copy) (a) Evaluate \(f(400)\) and interpret its meaning. (b) Graph \(f\) in the window \([0,600]\) by \([0,50] .\) Discuss how the elevation of the outer rail changes with the radius \(x\) (c) Interpret the horizontal asymptote. (d) What radius is associated with an elevation of 12.7 inches?

Solve each problem involving rate of work. Two grandparents want to pick up the mess that their granddaughter has made in her playroom. One can do it in 15 minutes working alone. The other, working alone, can clean it in 12 minutes. How long will it take them if they work together?

Solve each problem. Suppose \(p\) varies directly with the square of \(z\) and inversely with \(r .\) If \(p=\frac{32}{5}\) when \(z=4\) and \(r=10,\) find \(p\) when \(z=2\) and \(r=16\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.