/*! 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
The statement is true. The product of two even functions is also an even function.

Step by step solution

01

Understanding the Problem

Let's start by analyzing the definition of an even function. According to the mathematical definition, a function \(f(x)\) is called even if the following condition is met for all x in the function's domain: \(f(x) = f(-x)\). So, we have two such even functions f and g, and our goal is to prove that if f and g are even, then their compound function, that is, f times g, or simply \(fg\), is also even.
02

Going through the Proof

We will follow the direct proof approach. We start with the assumption that both \(f(x)\) and \(g(x)\) are even functions. To demonstrate that their product function \(f(x)g(x)\) is even, we need to prove that \((f g)(x) = (f g)(-x)\). Meaning, for any value of x, \((f g)(x)\) should be equal to \((f g)(-x)\). So let's start:From the premise that both f and g are even,\(f(x) = f(-x)\) and \(g(x) = g(-x)\).If we multiply these two equations, we get:\(f(x)g(x) = f(-x)g(-x)\).This implies \((fg)(x) = (fg)(-x)\), proving that the product of two even functions is also an even function.
03

Summarizing the Proof

The problem was to validate the statement 'If f and g are even functions, then their product fg is also an even function'. This was achieved by showing that if f and g are both even, i.e., \(f(x) = f(-x)\) and \(g(x) = g(-x)\), then their product function \(fg(x) = fg(-x)\), corresponds to the definition of an even function. Therefore, the original statement is valid.

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

If equations for \(f\) and \(g\) are given, explain how to find \(f-g\)

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation鈥檚 domain and range. $$(x+4)^{2}+(y+5)^{2}=36$$

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=4-x, g(x)=2 x^{2}+x+5$$

Solve and graph the solution set on a number line: $$3|2 x-1| \geq 21$$ (Section 1.7 , Example 10 )

Use a graphing utility to graph each circle whoseequation is given. Use a square setting for the viewing window. $$(y+1)^{2}=36-(x-3)^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.