/*! 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 8 Find the domain of the expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

Find the domain of the expression. $$-x^{4}+x^{3}+9 x$$

Short Answer

Expert verified
The domain of \(-x^4+x^3+9x\) is all real numbers \(\mathbb{R}\).

Step by step solution

01

Identify Expression Type

The expression is a polynomial: \(-x^{4}+x^{3}+9x\). Polynomials have the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).
02

Consider Domain Restrictions

Polynomials are defined for all real numbers because they involve only operations of addition, subtraction, multiplication, and non-negative integer exponents, which are defined everywhere.
03

State the Domain

Since there are no restrictions (like division by zero or square roots of negative numbers), the domain of a polynomial expression is all real numbers, denoted as \(\mathbb{R}\).

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
The domain of a function is the set of all possible input values (often represented as "x") that the function can accept. Not all functions can take every real number as input. For example, functions involving division by zero or square roots of negative numbers have limited domains. In straightforward terms, the domain tells us where the function is "allowed" to work without breaking any mathematical rules.

To find the domain, follow these steps:
  • Identify any restrictions, like division by zero or square roots of negative values.
  • If there are no operations that could cause issues, the domain is all real numbers, represented as \(\mathbb{R}\).
In the case of a polynomial, like \(-x^4+x^3+9x\), since it only involves multiplication and addition of reals, its domain is all real numbers.
Real Numbers
Real numbers include every number you can think of, except for complex numbers. These consist of integer numbers like -3, 0, 5, and decimal numbers like 1.5 and -2.8. Real numbers form a continuous line without any gaps, representing all the points along a line in geometry. They are denoted by \(\mathbb{R}\).

Why are real numbers important? They help us quantify real-world phenomena and everyday calculations. They form the basis for most of the arithmetic and algebraic structures we deal with in math.

You encounter real numbers daily, such as:
  • Counting items: "I have 3 apples" involves integers.
  • Reading a thermometer: "It's 23.7 degrees Celsius" involves decimals.
In polynomial expressions, every coefficient and result in a calculation are real numbers.
Polynomial Expression
A polynomial expression is a sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. Polynomials look like this: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

The degree of a polynomial is the highest power of the variable that appears with a non-zero coefficient. In our expression, \(-x^4 + x^3 + 9x\), the degree is 4 because the highest power is \(x^4\).

Some important features of polynomials:
  • They do not have variables in denominators or under square roots.
  • They are continuous and smooth curves when graphed.
  • Their domain is all real numbers, \(\mathbb{R}\), so they are always defined everywhere.
Understanding polynomials is crucial because they model a wide range of real-world problems and scenarios, from physics to economics.

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 gravitational force \(F\) exerted by the earth on an object having a mass of \(100 \mathrm{kg}\) is given by the equation $$F=\frac{4,000,000}{d^{2}}$$ where \(d\) is the distance (in \(\mathrm{km}\) ) of the object from the center of the earth, and the force \(F\) is measured in newtons (N). For what distances will the gravitational force exerted by the earth on this object be between \(0.0004 \mathrm{N}\) and \(0.01 \mathrm{N} ?\)

Gravity If an imaginary line segment is drawn between the centers of the earth and the moon, then the net gravitational force \(F\) acting on an object situated on this line segment is $$F=\frac{-K}{x^{2}}+\frac{0.012 K}{(239-x)^{2}}$$ where \(K>0\) is a constant and \(x\) is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the "dead spot" where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.) PICTURE CANT COPY

Inequalities Use the properties of inequalities to prove the following inequalities. Rule 6 for Inequalities: If \(a, b, c,\) and \(d\) are any real numbers such that \(a

Stopping Distance For a certain model of car the distance \(d\) required to stop the vehicle if it is traveling at \(v \mathrm{mi} / \mathrm{h}\) is given by the formula $$ d=v+\frac{v^{2}}{20} $$ where \(d\) is measured in feet. Kerry wants her stopping distance not to exceed 240 ft. At what range of speeds can she travel? (image cannot copy)

Simplify the expression. (a) \(\left(\frac{a^{1 / 6} b^{-3}}{x^{-1} y}\right)^{3}\left(\frac{x^{-2} b^{-1}}{a^{3 / 2} y^{1 / 3}}\right)\) (b) \(\frac{(9 s t)^{3 / 2}}{\left(27 s^{3} t^{-4}\right)^{2 / 3}}\left(\frac{3 s^{-2}}{4 t^{1 / 3}}\right)^{-1}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

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.