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Chapter 7: Problem 139

Find the domain \(D\) and the range \(R\) of the function \((x, x /|x|)\)

Short Answer

Expert verified
The domain of the function \((x, x/|x|)\) is given by \(D = (-\infty, 0) \cup (0, \infty)\), which includes all real numbers except 0. The range of the function is given by \(R = \{-1, 1\}\), where the function output can be either 1 or -1.

Step by step solution

01

Understand the given function

: We have a function \((x,x/|x|)\). The input of this function is \(x\), and the output is \(x/|x|\). This function is in the form of \((x,f(x))\), where \(f(x) = x/|x|\).
02

Find the domain D

: The domain encompasses all the possible values of \(x\) for which the function is defined. Since the function contains a division by the absolute value of \(x\), we need to make sure that the denominator is nonzero to find the valid values for \(x\). The absolute value of any non-zero number is always positive, so the only time the denominator will be zero is when \(x\) is 0. As a result, the domain of the function is the set of all real numbers except 0. In interval notation, the domain D is given by: \[D = (-\infty, 0) \cup (0, \infty)\]
03

Find the range R

: To find the range, we need to determine the possible values of the function output, which is given by \(x/|x|\). Let's examine the two cases for when \(x\) is positive and when \(x\) is negative: 1. If \(x>0\), then \(|x|=x\), and \(f(x) = \frac{x}{x} = 1\). 2. If \(x<0\), then \(|x|=-x\), and \(f(x) = \frac{x}{-x} = -1\). Since the function output can be either 1 or -1, the range R is given by: \[R = \{-1, 1\}\]

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Key Concepts

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

Absolute Value Function
The absolute value function is used to determine the non-negative magnitude of a number, regardless of its sign. It is represented by the notation \(|x|\), where \(x\) can be any real number. This function essentially "strips" a number of its sign, providing a positive output whether \(x\) is initially positive or negative. For instance, \(|3| = 3\) and \(|-3| = 3\).

The function in the exercise, \(x/|x|\), relies on the concept of absolute value. It divides the number \(x\) by its absolute value \(|x|\). This division helps determine the sign of \(x\) without considering its magnitude, as \(x/|x|\) results in either 1 or -1 depending on whether \(x\) is positive or negative.
Real Numbers
Real numbers include all the numbers you can think of, whether they are fractions, whole numbers, negatives, or decimals. They span the entire number line, covering everything from the tiniest decimal to the largest digit.

In the context of the exercise, \(x\) represents any real number except zero because division by zero is undefined. This means that \(x\) can be any value in the infinite set, both positive and negative, but cannot be zero. This wide range of input options is what makes the domain of our function so extensive, and specifically the set of all real numbers except zero.
Interval Notation
Interval notation is a way of describing specific portions of the number line where functions are defined or where solutions exist. It uses parentheses and brackets to show whether endpoints are included or excluded in an interval.

In this exercise, the domain of the function is expressed in interval notation as \((-\infty, 0) \cup (0, \infty)\). This indicates that \(x\) can be any real number except zero, combining two separate intervals: all numbers that are less than zero and all numbers greater than zero. The parentheses highlight that zero itself is not part of the domain.
Output Values
Output values, also known as the range, represent the set of possible outcomes of a function given its domain. For the function \(x/|x|\), determining the range involves evaluating the function for different signs of \(x\).

When \(x\) is positive, the absolute value is \(x\) itself, rendering the ratio \(x/|x| = 1\). Conversely, when \(x\) is negative, the absolute value is \(-x\), yielding \(x/|x| = -1\). Thus, the output values, or the range of the function, are limited to the discrete set \{-1, 1\}. These are the only possible results, regardless of the input, highlighting the dichotomy in outcomes based on the sign of \(x\).

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Most popular questions from this chapter

if \(\mathrm{f}(\mathrm{x})=(\mathrm{x}-2) /(\mathrm{x}+1)\), find the function values \(\mathrm{f}(2), \mathrm{f}(1 / 2)\) and \(\mathrm{f}(-3 / 4)\).

Find the zeros of the function \(\mathrm{f}\) if \(\mathrm{f}(\mathrm{x})=3 \mathrm{x}-5\).

Answer the following questions: (1) What is a linear function? (2) Give an example of a linear function and an example of a non-linear function. (3) Give an example of a function of 3 variables.

Draw a \(\begin{aligned}&\text { graph of the step function } \\\&\mathrm{y}= & 2 & -2 \leq \mathrm{x} \leq-1 \\\&\mathrm{y}= & 1 & -1 \leq \mathrm{x}<0 \\\&\mathrm{y}= & 0 & 0 \leq \mathrm{x}<1 \\\&\mathrm{y}= & 1 & 1 \leq \mathrm{x}<2 \\\&\mathrm{y}= & 2 & 2 \leq \mathrm{x} \leq 3\end{aligned}\)

Sketch the graph of the following functions: \(\begin{aligned}&\text { (a) } \mathrm{f}(\mathrm{x})=\left\\{\begin{array}{cc}\mathrm{x} \\ (1 / 2) \mathrm{x}-2\end{array}\right. & \begin{array}{l}\mathrm{x} \leq 4 \\\ \mathrm{x}>4\end{array} \\\&\text { (b) } \mathrm{f}(\mathrm{x})=|4 \mathrm{x}+9|\end{aligned}\) (c) \(\mathrm{f}(\mathrm{x})=|\mathrm{x}| /(\mathrm{x}+1)\) \(x \neq-1\)
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