/*! 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 41 Find the domain of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 41

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

Short Answer

Expert verified
The domain of \(f(x)=|x-4|\) is all real numbers, \((-\infty, \infty)\).

Step by step solution

01

Understand the Absolute Value Function

The absolute value function, denoted as \(|a|\), returns the non-negative value of \(a\). Hence, it does not affect the domain, because it is defined for all real numbers.
02

Define the Domain Conditions

To find the domain of \(f(x) = |x-4|\), note that there are no restrictions like division by zero or even roots of negative numbers. The absolute value function is defined for every real number.
03

State the Domain

Since there are no restrictions, the domain of \(f(x) = |x-4|\) is all real numbers. In set notation, this is written as \( (-\infty, \infty) \).

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.

Understanding the Absolute Value Function
The absolute value function is a crucial mathematical concept depicted as \(|a|\), where "a" can be any real number. This function essentially "strips" the sign from the number, resulting in its non-negative counterpart. For example:
  • \(|5| = 5\)
  • \(|-3| = 3\)
No matter what value "a" takes, the absolute value function will always deliver a non-negative result.

This characteristic implies that the function is widely applicable across all real numbers, without any extra conditions or exceptions.

When dealing with functions including absolute value, like \(f(x) = |x - 4|\), it's crucial to understand that the absolute value allows the function to encompass all real numbers as its domain.
Exploring Real Numbers
Real numbers form a vast and inclusive set that encompasses nearly all conceivable numbers encountered in typical mathematical computations. They include:
  • Whole numbers, both positive and negative, such as 7 or -2.
  • Fractions and decimals, like \(0.75\) or \(-\frac{3}{4}\).
  • Even irrational numbers like \(\pi\) or \(\sqrt{2}\).
A crucial feature of real numbers is their continuous nature on the number line, meaning there's always another real number between any two given real numbers.

This continuity aligns perfectly with the domain of functions like \(f(x) = |x - 4|\), which is defined across all possible real numbers, without restrictions.

Hence, when considering the domain of such functions, understanding the nature of real numbers is essential.
Using Set Notation for Domain Representation
Set notation is a powerful tool used in mathematics to specify and clarify the domain of functions concisely. When we say the domain is all real numbers, we denote it in set notation as \((-\infty, \infty)\). This indicates that there is no end to the range of real numbers the function can accept.
  • The round brackets \(()\) signify that the endpoints \(-\infty\) and \(\infty\) are not included, which is typical because infinity is a concept, not an actual number.
  • The expression \((-\infty, \infty)\) succinctly and neatly summarizes the function's capacity to handle any real number input.
This set notation not only streamlines mathematical communication but also avoids ambiguity, providing a clear understanding of the domain for functions like \(f(x) = |x - 4|\), which seamlessly incorporate all real numbers.

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 Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(y=x-150\)

The number of tickets sold for a fund-raiser is inversely proportional to the price of a ticket, \(p\). If 175 tickets can be sold for \(\$ 20\) each, how many tickets will be sold if the price is \(\$ 25\) each?

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 $$ A(t)=0.08 t+19.7 $$, 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 \(A(t)\)

Simplify. $$ 8^{2 / 3} $$

The Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(x=8-y^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics 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.