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Chapter 1: Problem 44

Find the domain of each function. Write your answer in interval notation. $$g(x)=-x^{3}-2$$

Short Answer

Expert verified
The domain of the function \(g(x)=-x^{3}-2\) is \((-鈭�, 鈭�)\)

Step by step solution

01

Understand the function

For the function \(g(x)=-x^{3}-2\), there is no restriction on the values of \(x\). This is because cubes and constants do not have any restrictions in the real number system.
02

Determine domain

Since the function does not give any restrictions for \(x\) value and it is defined for all real numbers, the domain is all real numbers. The domain in interval notation would be \((-鈭�, 鈭�)\)
03

Write the final answer

So, the domain of the function \(g(x)=-x^{3}-2\) is \((-鈭�, 鈭�)\)

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Key Concepts

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

Function Notation
Function notation is a way to represent functions in mathematics, which helps us understand how values are mapped to outputs. It is typically expressed in the form of "function name" followed by parentheses, like \( f(x) \). For example, in the function \( g(x) = -x^3 - 2 \), the letter "g" represents the name of the function, and "x" represents the input value or variable.

This notation enhances clarity when handling functions, making it easier to substitute and evaluate different inputs. When working with function notation:
  • Identify the function name, which specifies the rule being applied ("g" in the example).
  • Recognize the variable within the parentheses, indicating where you will substitute values ("x" in \( g(x) \)).
  • Understand the expression to the right of the equals sign to know what operations to perform with the input value.
Mastering function notation is key to advancing in algebra and calculus.
Interval Notation
Interval notation is a compact way to express the set of numbers that make up the domain or range of a function. It often uses brackets and parentheses to describe continuous sets of numbers on a number line.

In interval notation:
  • Parentheses \( () \) are used to describe intervals that do not include their endpoints. This means that the numbers at the ends of an interval are not part of the set. A common example is \((-\infty, \infty)\), which represents all real numbers.
  • Brackets \([\)] are used when the endpoints are included in the interval. For example, \([1, 5)\) includes all numbers from 1 to 5, but not the number 5 itself.
Interval notation is a powerful tool that enables a succinct representation of sets of numbers, especially when dealing with vast or infinite sets.
Real Numbers
Real numbers encompass all the numbers along the number line, from negative infinity to positive infinity. This set includes integers, fractions, and decimal numbers, essentially encompassing any value that can have a decimal representation.

The real numbers incorporate:
  • Natural numbers: \(1, 2, 3, \ldots \)
  • Whole numbers: \(0, 1, 2, 3, \ldots \)
  • Integers: \(\ldots, -2, -1, 0, 1, 2, \ldots \)
  • Rational numbers: Fractions and decimals that can be expressed as \(\frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\).
  • Irrational numbers: Numbers that cannot be expressed as simple fractions, such as \(\pi\) and \(\sqrt{2}\).
Understanding real numbers is fundamental to grasping advanced mathematical concepts, as they form the basis for calculus and many other branches of mathematics.

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Most popular questions from this chapter

A jogger on a pre-set treadmill burns 3.2 calories per minute. How long must she jog to burn at least 200 calories?

A long-distance telephone company advertises that it charges 1.00 dollars for the first 20 minutes of phone use and 7 cents a minute for every minute beyond the first 20 minutes. Let \(C(t)\) denote the total cost of a telephone call lasting \(t\) minutes. Assume that the minutes are nonnegative integers. (a) Many people will assume that it will cost only 0.50 dollars to talk for 10 minutes. Why is this incorrect? (b) Write an expression for the function \(C(t).\) (c) How much will it cost to talk for 5 minutes? 20 minutes? 30 minutes?

Sketch by hand the graph of the line with slope \(\frac{2}{5}\) that passes through the point \((1,-3) .\) Find the equation of this line.

This set of exercises will draw on the ideas presented in this section and your general math background. Find the intersection of the lines \(x+y=2\) and \(x-y=1\) You will have to first solve for \(y\) in both equations and then use the methods presented in this section. (This is an example of a system of linear equations, a topic that will be explored in greater detail in a later chapter.)

The sales of portable CD players in the United States held steady in the years 2000 and 2001 , but then slowly declined in the years \(2002-2005\) (Source: Consumer Electronics Association). The sales are given by the function $$S(t)=\left\\{\begin{array}{ll}32, & \text { if } 0 \leq t \leq 1 \\\32-3.75(t-1), & \text { if } 1
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