/*! 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 47 Find the domain of the function.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 47

Find the domain of the function. Write the domain using interval notation. $$N(x)=\log _{2}\left(x^{3}-x\right)$$

Short Answer

Expert verified
The domain of the function \(N(x)=\log_{2}\left(x^{3}-x\right)\) in the interval notation is \((-鈭�, -1) 鈭� (0, 1) 鈭� (1, 鈭�)\).

Step by step solution

01

Set the Equation

Find out where the argument of the log function is greater than zero. Solve the inequality \(x^{3}-x > 0\).
02

Factor the Polynomial

Factor the left side of the inequality to more easily find its roots. The factored form of \(x^{3}-x\) is \(x(x^2-1)\) or \(x(x-1)(x+1)\).
03

Solve the Inequality

The roots of the inequality \(x(x-1)(x+1)>0\) are -1, 0, and 1. We create a sign chart with these points as endpoints and determine the sign of the function in the intervals. From this, we can infer that \(x< -1\), \(0 < x < 1\) and \(1 < x\).

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.

Polynomial Inequalities
Polynomial inequalities are expressions involving polynomials that we want to compare. In our case, we have an inequality of the form \(x^3 - x > 0\). Solving these inequalities means figuring out over which intervals of the variable, here \(x\), the polynomial takes a positive value. Let's break down the process:
  • First, factor the polynomial. Our polynomial is \(x^3 - x\), which simplifies to \(x(x^2 - 1)\). Further, \(x^2 - 1\) is a difference of squares that can be factorized into \((x - 1)(x + 1)\). Thus, we get \(x(x - 1)(x + 1)\).
  • Next, find the roots by setting each factor equal to zero. This gives us \(x = -1\), \(x = 0\), and \(x = 1\).
  • After determining the roots, we test the sign of the inequality in the intervals defined by these roots. We choose test points in these intervals like \(x = -2\) for \(x < -1\), \(x = -0.5\) for \(0 < x < 1\), etc., and substitute them back into the inequality to see if the expression is positive or negative in those intervals.
  • The polynomial will be positive in the intervals where our inequality holds true, which we find to be \((-\infty, -1)\), \((0, 1)\), and \((1, \infty)\).
Polynomial inequalities often appear in calculus and algebra, helping define regions of interest on number lines where functions behave in certain ways.
Interval Notation
Interval notation is a way to represent subsets of the real number line. It is especially useful to express solutions to inequalities and domains. Let's consider the interval notation relevant to the problem:
  • With the inequality \(x^3 - x > 0\), we deduced the solution intervals: \((-\infty, -1)\), \((0, 1)\), and \((1, \infty)\). These intervals tell us where \(x^3 - x\) is positive.
  • Each piece of interval notation includes numbers and symbols. For example, \((a, b)\) represents all numbers between \(a\) and \(b\), not including \(a\) or \(b\). Square brackets \([a, b]\) would mean \(a\) and \(b\) are included.
  • The symbol \(-\infty\) or \(\infty\) in interval notation reflects an interval that goes on indefinitely in the negative or positive direction. Infinity is never included within the interval since it's not a real number, hence the use of parentheses.
This notation is concise and efficient, allowing for precise communication of the domain of a function or the solution set of an inequality. For instance, the domain of our original exercise's function \(N(x)=\log _{2}(x^{3}-x)\) in interval notation is expressed as \((-\infty, -1) \cup (0, 1) \cup (1, \infty)\). This tells us exactly where the function exists in terms of the possible input values \(x\).
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions and are defined for positive arguments only. In mathematical terms, if \(y = \log_b(x)\), this means \(b^y = x\), where \(b\) is the base of the logarithm. In our exercise, we're working with a specific logarithmic function: \(N(x) = \log_2(x^3 - x)\).
  • The domain of a logarithmic function consists of all \(x\) values for which the argument \(x^3 - x\) is greater than zero (\(x^3 - x > 0\)). This is because the logarithm is undefined for zero or negative numbers.
  • For a base like 2, as in \(\log_2\), this refers to the logarithm of a number to the base 2. The value of the logarithm is the power to which this base must be raised to obtain the argument, \(x^3 - x\), in our case.
  • Understanding the domain is crucial when working with logarithmic functions because it defines the inputs, \(x\), that ensure the function remains valid and meaningful.
Recognizing the characteristics of logarithmic functions and how they relate to their domains can greatly aid in solving problems related to these functions, ensuring both clarity and correctness.

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

A model for how long our aluminum resources will last is given by $$T=\frac{\ln (20,500 r+1)}{\ln (r+1)}$$ where \(r\) is the percent increase in consumption from current levels of use and \(T\) is the time (in years) before the resource is depleted. a. Use a graphing utility to graph this equation. b. If our consumption of aluminum increases by \(5 \%\) per year, in how many years (to the nearest year) will we deplete our aluminum resources? c. What percent increase in consumption of aluminum will deplete the resource in 100 years? Round to the nearest tenth of a percent.

Explain how to use the graph of the first function \(f\) to produce the graph of the second function \(F\). $$f(x)=e^{x}, F(x)=e^{-x}+2$$

Assuming that air resistance is proportional to velocity, the velocity \(v,\) in feet per second, of a falling object after \(t\) seconds is given by \(v=32\left(1-e^{-t}\right)\) a. Graph this equation for \(t \geq 0\) b. Determine algebraically, to the nearest 0.01 second, when the velocity is 20 feet per second. c. Determine the horizontal asymptote of the graph of \(v\) d. the horizontal asymptote in the context of this application.

The percent \(I(x)\) of the original intensity of light striking the surface of a lake that is available \(x\) feet below the surface of the lake is given by \(I(x)=100 e^{-0.95 x}\) a. What percentage of the light, to the nearest tenth of a percent, is available 2 feet below the surface of the lake? b. At what depth, to the nearest hundredth of a foot, is the intensity of the light one-half the intensity at the surface?

Assuming that air resistance is proportional to velocity, the velocity \(v,\) in feet per second, of a falling object after \(t\) seconds is given by \(v=64\left(1-e^{-t / 2}\right)\) a. Graph this equation for \(t \geq 0\) b. Determine algebraically, to the nearest 0.1 second, when the velocity is 50 feet per second. c. Determine the horizontal asymptote of the graph of \(v\). d. Write a sentence that explains the meaning of the horizontal asymptote in the context of this application.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.