/*! 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 73 Prove that the product of two ev... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 73

Prove that the product of two even (or two odd) functions is even.

Short Answer

Expert verified
The product of two even functions or two odd functions is always an even function.

Step by step solution

01

Define the functions

Let's consider two even functions, \( f(x) \) and \( g(x) \), such that \( f(-x) = f(x) \) and \( g(-x) = g(x) \). We will prove this for even functions first, and the proof for odd functions will follow from the definition of odd functions.
02

Express the product function

The product of functions \( f(x) \) and \( g(x) \) is \( h(x) = f(x)g(x) \). Our goal is to prove that \( h(x) \) is also an even function.
03

Substitute and simplify

To show that \( h(x) \) is an even function, we need to demonstrate that \( h(-x) = h(x) \). So, we substitute -x into our product function: \( h(-x) = f(-x)g(-x) \). Since \( f(-x) = f(x) \) and \( g(-x) = g(x) \), then \( h(-x) = f(x)g(x) = h(x) \). Thus, the product of two even functions is also an even function.
04

Repeat for odd functions

For two odd functions \( p(x) \) and \( q(x) \), they satisfy \( p(-x) = -p(x) \) and \( q(-x) = -q(x) \). The product of these functions, \( r(x) = p(x)q(x) \), would yield \( r(-x) = -p(x)(-q(x)) = p(x)q(x) = r(x) \). Thus, the product of two odd functions is an even function also.

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 \(f(x)=\sin x \quad\) and \(\quad g(x)=\arcsin (\sin x)\) Why isn't the graph of \(g\) the line \(y=x ?\)

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 3} \frac{x-2}{x^{2}} $$

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=\frac{x^{2}+x}{x-1}, \quad\left[\frac{5}{2}, 4\right], \quad f(c)=6 $$

Describe how the functions \(f(x)=3+\llbracket x \rrbracket\) and \(g(x)=3-\llbracket-x \rrbracket\) differ.

Solve the equation for \(x\). $$ \arccos x=\operatorname{arcsec} x $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.