/*! 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 72 Graph the functions \(f\) and \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 72

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\cos ^{2} \frac{\pi x}{2}, \quad g(x)=\frac{1}{2}(1+\cos \pi x)$$

Short Answer

Expert verified
The conjecture about the relationship between \(f(x) = \cos ^{2} \frac{\pi x}{2}\) and \(g(x) = \frac{1}{2}(1+\cos \pi x)\) is that these two functions are equivalent, i.e., \(f(x) = g(x)\) for all \(x\).

Step by step solution

01

Graph the function \(f(x)=\cos ^{2} \frac{\pi x}{2}\)

Start by setting up your graph for \(f\). Since \(f(x)\) involves a cosine function multiplied by a constant and a variable, it will have a wave-pattern. The period of this wave will be determined by the denominator of \(\pi x\) in the cosine function. In this case, the period will be 2. The cosine function here is squared, so the graph is always positive. Draw a cosine wave with a period of 2, and all values positive.
02

Graph the function \(g(x)=\frac{1}{2}(1+\cos \pi x)\)

Next, set up the graph for function \(g\). This function is a shifted and scaled cosine wave. The cosine wave is shifted up by 1, indicated by the \(1+\) preceding the cosine function, and the entire function is scaled by a factor of \(1/2\), indicated by the multiplier outside the parentheses. The period is 1, as indicated by the \(\pi x\) inside the cosine function. The graph will therefore look like a regular cosine wave with a period of 1, shifted upwards by 1 unit and halved in amplitude.
03

Analyze the graphs

With both graphs drawn, compare the two. It should become evident that the two functions produce identical graphs - they are the same wave, just expressed in different mathematical forms. Therefore, the conjecture is that \(f(x) = g(x)\) for all \(x\), given that the two functions produce the same graphical output.

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

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\) (a) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\sec x$$

True or False? Determine whether the statement is true or false. Justify your answer. The difference between the measures of two coterminal angles is always a multiple of \(360^{\circ}\) when expressed in degrees and is always a multiple of \(2 \pi\) radians when expressed in radians.

A ship leaves port at noon and has a bearing of \(\mathrm{S} 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6: 00 P.M.? (b) At 6: 00 e.m., the ship changes course to due west. Find the ship's bearing and distance from the port of departure at 7: 00 P.M.

Determine whether the statement is true or false. Justify your answer. You can obtain the graph of \(y=\sec x\) on a calculator by graphing a translation of the reciprocal of \(y=\sin x\)

A car's rear windshield wiper rotates \(125^{\circ} .\) The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.