/*! 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 70 Find the only function whose dom... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 70

Find the only function whose domain is the set of real numbers and is both even and odd.

Short Answer

Expert verified
The function that has the domain of all real numbers and is both even and odd is \(f(x) = 0\).

Step by step solution

01

Define Even and Odd Functions

An even function is a function that satisfies the condition f(-x) = f(x), and an odd function is a function that satisfies the condition f(-x) = -f(x).
02

Set Up the Equation

Since the given function is both even and odd, we can set up the following equation using the function f such that: f(-x) = f(x) = -f(x)
03

Combine the Two Conditions

Let's find f(x) by combining the equations for an even and an odd function: f(x) = -f(x)
04

Solve for f(x)

In order to find the function f(x), we can add f(x) to both sides of the equation: f(x) + f(x) = 0 2f(x) = 0 Now, to solve for f(x), we can simply divide both sides of the equation by 2: f(x) = 0
05

Final Answer

The function we are looking for, which is both even and odd and has the domain of all real numbers, is: f(x) = 0

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function
Understanding the domain of a function is essential as it represents all the possible inputs for the function. Specifically, the domain is the set of all values that can be plugged into a function without causing any undefined situations, such as division by zero or taking the square root of a negative number. For example, a function like \( f(x) = \sqrt{x} \) has a domain of all non-negative real numbers, because square roots of negative numbers are not defined in the set of real numbers.
Defining Even Function
An even function is symmetric with respect to the y-axis, which means that its graph is identical on both sides of the y-axis. Mathematically, a function \( f \) is even if for every number \( x \) in the domain of \( f \), \( f(-x) = f(x) \) holds true. This property results in symmetric graphs and can be useful in simplifying integrals and solving problems involving symmetry. A classic example of an even function is \( f(x) = x^2 \), where the squares of \( x \) and \( -x \) are always equal.
Defining Odd Function
Conversely, an odd function has rotational symmetry around the origin, which means if its graph is rotated 180 degrees about the origin, it will look the same. A function \( f \) is odd if, for every \( x \) in its domain, \( f(-x) = -f(x) \) is valid. This reflects in graphs where flipping the graph across the origin (both the x and y-axis) will yield the same graph. Many trigonometric functions, like \( f(x) = \sin(x) \), are odd functions and exhibit this behavior.
Properties of Even and Odd Functions
Even and odd functions have distinct properties that can be utilized in various mathematical analyses. For even functions, integrals over symmetric bounds from \( -a \) to \( a \) will result in double the integral from 0 to \( a \) due to their symmetry. On the other hand, the integral of an odd function over symmetric limits is zero, since the areas under the curve from \( -a \) to \( 0 \) and from 0 to \( a \) cancel out. Furthermore, the only function that is both even and odd is the constant function \( f(x) = 0 \), as any non-zero values would violate either the even or odd function properties. This is a unique case and an interesting aspect of mathematical symmetry.

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

Find a number b such that the function \(f\) equals the function \(g\). Both \(f\) and \(g\) have domain \\{3,5\\} , with \(f\) defined on this domain by the formula \(f(x)=x^{2}-3\) and \(g\) defined on this domain by the formula \(g(x)=\frac{18}{x}+b(x-3)\)

Suppose \(f(x)=x^{3}+x\). Which of the following points is on the graph of \(f^{-1}\) ? $$ (4,1), \quad(2,10), \quad(10,2) $$

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

Sketch the graph of a function whose domain is the interval [1,3] and whose range is the interval [-2,4] .

A formula has been given defining a function \(f\) but no domain has been specified. Find the domain of each function \(f\), assuming that the domain is the set of real numbers for which the formula makes sense and produces \(a\) real number. \(f(x)=\frac{\sqrt{x-5}}{x-7}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

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.