/*! 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 113 a. Graph the functions \(f(x)=x^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 113

a. Graph the functions \(f(x)=x^{n}\) for \(n=2,4,\) and 6 in a \([-2,2,1]\) by \([-1,3,1]\) viewing rectangle. b. Graph the functions \(f(x)=x^{n}\) for \(n=1,3,\) and 5 in a \([-2,2,1]\) by \([-2,2,1]\) viewing rectangle. c. If \(n\) is positive and even, where is the graph of \(f(x)=x^{n}\) increasing and where is it decreasing? d. If \(n\) is positive and odd, what can you conclude about the graph of \(f(x)=x^{n}\) in terms of increasing or decreasing behavior? e. Graph all six functions in a \([-1,3,1]\) by \([-1,3,1]\) viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?

Short Answer

Expert verified
a. The graphs of \(f(x)=x^{2}\), \(f(x)=x^{4}\) and \(f(x)=x^{6}\) within the viewing rectangle [-2, 2, 1] by [-1, 3, 1] are U-shaped. b. the graphs of \(f(x)=x^{1}\), \(f(x)=x^{3}\) and \(f(x)=x^{5}\) within the viewing rectangle [-2, 2, 1] by [-2, 2, 1] are increasing from left to right. c. For a positive and even \(n\), \(f(x)=x^{n}\) is decreasing for \(x < 0\), and increasing for \(x > 0\). d. For a positive and odd \(n\), \(f(x)=x^{n}\) is always increasing. e. The graph becomes steeper as the value of n increases. For negative x-values, odd power functions decrease steeply, whereas even power functions flatten.

Step by step solution

01

Graphing even functions

Plot the functions \(f(x) = x^{2}\), \(f(x) = x^{4}\) and \(f(x) = x^{6}\) within the viewing rectangle [-2, 2, 1] by [-1, 3, 1].
02

Graphing odd functions

Plot the functions \(f(x) = x^{1}\), \(f(x) = x^{3}\), and \(f(x) = x^{5}\) within the viewing rectangle [-2, 2, 1] by [-2, 2, 1].
03

Analyzing increasing and decreasing behavior for even functions

For a positive and even \(n\), \(f(x) = x^{n}\) is decreasing for \(x < 0\), and increasing for \(x > 0\). At \(x = 0\), we have minimum points, which can be observed in the graphs of step 1 as well.
04

Analyzing increasing and decreasing behavior for odd functions

For a positive and odd \(n\), \(f(x) = x^{n}\) is always increasing. This can be observed in the graphs of step 2.
05

Graphing all six functions

Plot all six functions \(f(x) = x^{n}\), where n is 1, 2, 3, 4, 5, and 6 within the viewing rectangle [-1, 3, 1] by [-1, 3, 1] as in step 1 and step 2.
06

Analyzing the graphs

From the graphs in step 5, it can be seen that as the value of n increases, the graph becomes steeper. For negative x-values, the odd power functions tend to decrease steeply, whereas even power functions tend to flatten.

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!

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

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x-2)^{2}+(y-3)^{2}=16 $$

Exercises \(129-131\) will help you prepare for the material covered in the next section. The function \(C(t)=20+0.40(t-60)\) describes the monthly cost, \(C(t),\) in dollars, for a cellphone plan for \(t\) calling minutes, where \(t>60 .\) Find and interpret \(C(100)\)

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ 0=-2(x-3)^{2}+8 $$

The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve Exercises \(101-102\) a. Find and interpret \(f(70)\) b. Find and interpret \(g(70)\) c. Which function serves as a better model for the chance of surviving to age \(70 ?\)

In Exercises \(105-108,\) you will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used, \(x .\) Then find and interpret \(S(30)\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.