/*! 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 118 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 118

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.

Short Answer

Expert verified
True, the product of two even functions is also an even function as established by the proof.

Step by step solution

01

Define the functions

Let \(f\) and \(g\) be even functions, so it is true that \(f(-x) = f(x)\) and \(g(-x) = g(x)\) for all \(x\).
02

Consider the Product Function

We define \(h(x) = f(x)g(x)\). We need to prove that \(h(-x) = h(x)\) for all \(x\).
03

Prove the Product Function is Even

Evaluate \(h(-x)\) by substituting \(-x\) into the definition of \(h\): \[h(-x) = f(-x)g(-x).\] Since \(f\) and \(g\) are both even functions, this is equal to \[f(x)g(x) = h(x),\] which proves that \(h\) is even. Thus, the product of two even functions is also an even function.

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

What is the horizontal line test and what does it indicate?

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=(x-1)^{3} $$

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac 12 x^{3} $$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt[3]{x-2} $$

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$h(x)=2-x^{\frac{2}{5}}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.