/*! 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 30 Decide whether the function is e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 30

Decide whether the function is even, odd, or neither. \(h(x)=x^{3}+3\)

Short Answer

Expert verified
The function \(h(x) = x^{3} + 3\) is neither even nor odd.

Step by step solution

01

Define the function

Firstly, write down the function that you are given, which is \(h(x) = x^{3} + 3\).
02

Evaluate h(-x)

Now, substitute \(-x\) for \(x\) in the function to get \(h(-x)\). Function then becomes \(h(-x) = (-x)^{3} + 3 = -x^{3} + 3\).
03

Compare with h(x) and -h(x)

Then, compare \(h(-x)\) with \(h(x)\) and \(-h(x)\). The function \(h(-x) = -x^{3} + 3\) is not the same as \(h(x) = x^{3} + 3\), therefore the function is not even. Also, \(h(-x) = -x^{3} + 3\) is not equal to \(-h(x) = -(x^{3} + 3) = -x^{3} - 3\), therefore the function is not odd.
04

Conclude the result

Since \(h(-x)\) does not match either \(h(x)\) or \(-h(x)\), we can deduce that the function \(h(x) = x^{3} + 3\) is neither even nor odd.

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 examining the nature of a function, understanding its symmetry is crucial in determining whether it's even, odd, or neither. Symmetry simplifies visualizing mathematical behaviors, but first, let's break down what these terms mean.

  • Even Functions: A function is even if it satisfies the condition \(f(-x) = f(x)\) for all values of \(x\). Graphically, even functions are symmetric about the vertical y-axis. Common examples include trigonometric functions like cosine.
  • Odd Functions: A function is odd if \(f(-x) = -f(x)\) for all \(x\). In terms of symmetry, odd functions are symmetric about the origin, meaning a 180-degree rotation returns the graph to its original position. Sine and tangent are typical odd functions.
  • Neither: Functions that do not satisfy either condition are classified as neither even nor odd, lacking any specific symmetry.
For example, the function \(h(x) = x^3 + 3\) doesn't fit either criterion because \(h(-x) = -x^3 + 3\) isn't equal to \(h(x)\) or \(-h(x)\), showing it's neither even nor odd.
Algebraic Functions
Algebraic functions are composed of basic algebraic operations: addition, subtraction, multiplication, division, or raising to a power. They are often presented as polynomials, radical functions, or quotients of polynomials.

Algebraic functions have some defining traits:
  • Standard Form: Generally, they take the form of \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are constants.
  • Simplicity: They involve polynomial expressions and rational expressions, making them more straightforward to work with compared to transcendental functions (like logarithmic or exponential).
In our example, \(h(x) = x^3 + 3\), we see a simple algebraic expression where a cubic term is combined with a constant. Understanding these functions helps lay the groundwork for exploring broader mathematical concepts.
Polynomial Functions
Polynomial functions, a vital subclass of algebraic functions, form the backbone of many algebraic applications. They include terms with whole number exponents. Let's explore what makes them special.

  • Structure: A polynomial is expressed as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n eq 0\) and \(n\) is a non-negative integer. Each term consists of a coefficient and a power of \(x\).
  • Degree: Defined by the highest power of \(x\) in the function. For \(h(x) = x^3 + 3\), the degree is 3, making it a cubic polynomial. Higher degrees generally mean more complexity.
  • Behavior: Polynomial functions are continuous and smooth, with no gaps or sharp corners, making them predictable and easy to analyze.
In essence, polynomial functions provide a foundation for understanding complex relationships through their straightforward structure, as seen in the given function \(h(x) = x^3 + 3\). This specific polynomial is straightforward, yet demonstrates the key features that define polynomial behavior.

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) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=\sqrt{x^{2}-4}, \quad g(x)=\frac{x^{2}}{x^{2}+1}\)

A unPower Corporation develops and manuAactures solar-electric power products. SunPower's new higher efficiency solar cells generate up to \(50 \%\) more power than other solar technologies. SunPower's technology was developed by Dr. Richard Swanson and his students while he was Professor of Engineering at Stanford University. SunPower's 2006 revenues are projected to increase \(300 \%\) from its 2005 revenues. Use your campus library, the Internet, or some other reference source to find information about an alternative energy business experiencing strong growth similar to the example above. Write a brief report about the company or small business.

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=2 x-3, \quad g(x)=2 x-3\)

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is shifted three units to the left.

Find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(f \circ f\). \(f(x)=x^{3}, \quad g(x)=\frac{1}{x}\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.