/*! 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 42 Find the domain of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 42

Find the domain of each function given below. $$f(x)=|x|-4$$

Short Answer

Expert verified
The domain is \( (-\infty, \infty) \).

Step by step solution

01

Understand the Function

The function given is a piecewise linear function involving the absolute value of the variable x minus 4, i.e., \( f(x) = |x| - 4 \). The absolute value function is defined for all real numbers.
02

Determine The Domain

Identify the set of all possible input values (x-values) for the function. Since there are no restrictions such as division by zero or even roots of negative numbers, the function \( f(x) = |x| - 4 \) is defined for all real numbers.
03

Express the Domain

To express the domain in interval notation, note that the function can take any real number as input. Therefore, the domain is \( (-\infty, \infty) \).

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.

Absolute Value Function
The absolute value function is a fundamental concept in mathematics. It is denoted by the symbol \( |x| \). This function returns the non-negative value of any real number x.
For example: \( |3| = 3 \) and \( |-3| = 3 \).
Unlike most functions, the absolute value function does not change whether x is positive or negative.
Here are key points to remember:
  • It always outputs non-negative values.
  • It is symmetrical around the y-axis because \( | -x | = | x | \).
In our example, \( f(x) = |x| - 4 \), the output value is the absolute value of x minus 4. Since we are only dealing with the absolute value, it simplifies understanding of function behavior.
Real Numbers
Real numbers include all the numbers we typically use in daily life.
They encompass:
  • Positive numbers like 1, 2, 3,
  • Negative numbers like -1, -2, -3,
  • Zero,
  • Fractions like 1/2, 2/3,
  • Decimals like 0.1, -0.5.
All these fit within the set of real numbers.
Real numbers are denoted by the symbol \( \mathbb{R} \).
They also have useful properties:
  • They can be ordered on a number line.
  • They are continuous, meaning there are no gaps between them.
  • They can be found in every part of daily life, from measuring distances to calculating money.
In our function \( f(x) = |x| - 4 \), the variable x represents any real number, which means there are no restrictions for x.
Interval Notation
Interval notation is a way to represent continuous sets of real numbers.
It is especially useful for describing domains of functions.
Instead of listing all individual numbers, we use intervals:
  • An interval like \( (a, b) \) represents all numbers between a and b, not including a and b.
  • An interval like \( [a, b] \) includes the numbers a and b.
  • For infinite ranges, we use \( \infty \).
  • \( (-\infty, \infty) \) means all numbers from negative infinity to positive infinity (the entire set of real numbers).
For our function \( f(x) = |x| - 4 \), the domain is all real numbers. In interval notation, we write this as \( (-\infty, \infty) \).
This compactly indicates that the function accepts any real number as an input.

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

Simplify. $$16^{5 / 2}$$

Use the vertical-line test to determine whether each graph is that of a function. (The vertical dashed lines are not part of the graph.)

For \(f(x)=x^{2}+4 x,\) find \(\frac{f(x+h)-f(x)}{h}.\)

For \(f(x)=x^{2}-3 x,\) find \(\frac{f(x+h)-f(x)}{h}.\)

In general, people in our society are marrying at a later age. The median age, \(A(t),\) of women at first marriage can be approximated by the linear function \(A(t)=0.08 t+19.7\) where \(t\) is the number of years after \(1950 .\) Thus, \(A(0)\) is the median age of women at first marriage in 1950 \(A(50)\) is the median age in \(2000,\) and so on. a) Find \(A(0), A(1), A(10), A(30),\) and \(A(50)\) b) What was the median age of women at first marriage in \(2008 ?\) c) Graph \(A(t)\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.