/*! 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 63 Show that the product of two eve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 63

Show that the product of two even functions (with the same domain) is an even function.

Short Answer

Expert verified
To prove that the product of two even functions is also an even function, let's denote the given functions as \(f(x)\) and \(g(x)\), both satisfying the condition \(f(x) = f(-x)\) and \(g(x) = g(-x)\) for all x in their domain. Let h(x) be the product of these functions, so \(h(x) = f(x) * g(x)\). We now substitute -x for x and get \(h(-x) = f(-x) * g(-x)\). Using the properties of even functions, we have \(h(-x) = f(x) * g(x)\), which means \(h(-x) = h(x)\), proving that the product function h(x) is also even.

Step by step solution

01

Define the given functions

Let's denote the two given even functions as f(x) and g(x), where f(x) = f(-x) and g(x) = g(-x) for all x in their domain.
02

Find the product of the functions

To reveal that the product of the functions is even as well, we first need to express their product. Let h(x) be their product, so h(x) = f(x) * g(x) for all x in their domain.
03

Substitute with the negative argument

Now, we will substitute x with -x in the expression for h(x), and show that h(-x) = h(x) for the product of f(x) and g(x). This will prove that theproduct is also an even function. h(-x) = f(-x) * g(-x)
04

Use the properties of even functions

Given that f(x) and g(x) are even functions, we can use their respective properties of evenness here: h(-x) = f(x) * g(x) This means that h(-x) = h(x) which implies that the product of the two even functions, h(x), is an even function as well.
05

Conclusion

We have demonstrated that the product of two even functions, f(x) and g(x), is also an even function, since h(-x) = h(x) for h(x) = f(x) * g(x). Therefore, the product of two even functions with the same domain is 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

Suppose the income tax function in Example 2 of Section 1.1 is changed so that $$g(x)=0.15 x-450 \quad \text { if } 8500

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Sketch the graph of \(f^{-1}\).

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Give the table of values for \(f^{-1} \circ g^{-1}\).

A temperature \(F\) degrees Fahrenheit corresponds to \(g(F)\) degrees on the Kelvin temperature scale, where $$g(F)=\frac{5}{9} F+255.37$$ (a) Find a formula for \(g^{-1}(K)\). (b) What is the meaning of \(g^{-1}(K) ?\) (c) Evaluate \(g^{-1}(0)\). (This is absolute zero, the lowest possible temperature, because all molecular activity stops at 0 degrees Kelvin.)

Show that the composition of two one-to-one functions is a one-to-one function.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.