/*! 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 37 Decide if each function is odd, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 37

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=-x^{3}+1$$

Short Answer

Expert verified
The function \(f(x) = -x^3 + 1\) is an odd function.

Step by step solution

01

Compute f(-x)

The first step is to substitute \(-x\) for \(x\) in the function. This gives \(f(-x) = -(-x)^3 + 1 = -x^3 + 1\).
02

Compare f(-x) with f(x) and -f(x)

We see that \(f(-x) = -x^3 + 1\) is not equal to \(f(x) = -x^3 + 1\), so the function cannot be even. However, we see that \(f(-x) = -f(x)\), which satisfies the condition for odd functions.
03

Determine the type of function

Since \(f(-x) = -f(x)\), the function \(f(x) = -x^3 + 1\) is an odd 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!

Key Concepts

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

Function Symmetry
When we talk about function symmetry, we primarily discuss even and odd functions. Function symmetry is essential in mathematics because it helps to understand the behavior and properties of a function.

**Even Functions**: These functions are symmetric with respect to the y-axis. For a function to be even, the following must hold true:
  • The condition \( f(-x) = f(x) \) applies for all x in the domain.
  • A classic example of an even function is \( f(x) = x^2 \).
**Odd Functions**: Odd functions, on the other hand, are symmetric with respect to the origin. This leads to a particular property:
  • For a function to be odd, \( f(-x) = -f(x) \) should be true for all x in the domain.
  • An example of an odd function is \( f(x) = x^3 \).
Understanding these symmetries can simplify graphing functions and solving equations. Try plotting both function types to visually see the symmetry!
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables. Each term in a polynomial function consists of a coefficient, a variable, and an exponent.
  • A polynomial is typically written in the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants.
  • In the simplest case, such as\( f(x) = -x^3 + 1 \), we see a polynomial of degree 3.
Polynomials can exhibit characteristics of odd or even functions depending on their constituent terms:
  • Polynomial functions with all even degree terms are usually even functions.
  • If all the terms have odd degrees, they're typically odd functions.
In our exercise, the function \( f(x) = -x^3 + 1 \) includes both odd and constant terms; however, its behavior skews towards being an odd function due to the highest-degree term being odd.
Mathematical Proofs
Understanding mathematical proofs is crucial when verifying the nature of a function's symmetry. Proofs give us a step-by-step logic that confirms whether a function is even, odd, or neither.

Let's examine the proof for odd functions, using our original function \( f(x) = -x^3 + 1 \) as an example:
  • We start by computing \( f(-x) \), replacing \( x \) with \( -x \), resulting in \( -(-x)^3 + 1 = -x^3 + 1 \).
  • Compare this with \( f(x) \). Here, you'll notice \( f(-x) = f(x) \) doesn't hold, indicating it's not even.
  • Next, observe that \( f(-x) = -(-x^3 + 1) = -f(x) \), satisfying the condition for odd functions.
By carefully following these proof steps, you can confidently determine a function's symmetry. This logical process reinforces mathematical understanding and validates your solution with certainty.

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

The surface area of a sphere is given by \(A(r)=4 \pi r^{2},\) where \(r\) is in inches and \(A(r)\) is in square inches. The function \(C(x)=6.4516 x\) takes \(x\) square inches as input and outputs the equivalent result in square centimeters. Find \((C \circ A)(r)\) and explain what it represents.

Let \(f(x)=2 x+5\) and \(g(x)=f(x+2)-4 .\) Graph both functions on the same set of coordinate axes. Describe the transformation from \(f(x)\) to \(g(x) .\) What do you observe?

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$f(t)=-t^{2}-1$$

The following table gives the average hotel room rate for selected years from 1990 to \(1999 .\) (Source:American Hotel and Motel Association) $$\begin{array}{cc}\text { Year } & \text { Rate (in dollars) } \\\\\hline 1990 & 57.96 \\\1992 & 58.91 \\\1994 & 62.86 \\\1996 & 70.93 \\\1998 & 78.62 \\\1999 & 81.33\end{array}$$ (a) What general trend do you notice in these figures? (b) Fit both a linear and a quadratic function to this set of points, using the number of years since 1990 as the independent variable. (c) Based on your answer to part (b), which function would you use to model this set of data, and why? (d) Using the quadratic model, find the year in which the average hotel room rate will be \(\$ 85\)

In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=\sqrt[3]{4 x^{2}-1}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics 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.