/*! 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 66 Find the domain of the function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 66

Find the domain of the function \(f(x)=\frac{1}{\sqrt{4 x-\left\lfloor x^{2}-10 x+9 \mid\right.}}\)

Short Answer

Expert verified
The function does not have a domain in the real number system.

Step by step solution

01

Solve the inequality

As we have stated in analysis, we need to find the roots of the function \(x^2 - 10x + 9\). Those roots will be used to split the number line and check the solutions of the inequality \(4x - |x^2 - 10x + 9| > 0\). You can find the roots by Factoring or using the Quadratic equation formula, which is \((-b \pm \sqrt{b^2 - 4ac})/2a\). After calculation, the roots will be 1 and 9.
02

Split the inequality

Use the roots to split the inequality into three regions: Region 1 (\(x < 1\)), Region 2 (\(1 \leq x \leq 9\)), and Region 3 (\(x > 9\)). Now the inequality \(4x - |x^2 - 10x + 9| > 0\) can be rewritten as \(4x - (x^2 - 10x + 9) > 0\) for Region 1 and Region 3, and \(4x + (x^2 - 10x + 9) > 0\) for Region 2.
03

Solve each splitted inequality

For Region 1 and Region 3, the simplified inequality is \(-x^2 + 14x - 9 > 0\). Collecting terms gives you \(x^2 - 14x + 9 < 0\). This equation doesn't have any real roots, hence, there are no solutions in Region 1 and Region 3. For Region 2, the simplified inequality is \(x^2 - 6x + 9 < 0\). The left-hand side of this inequality can be factored as \((x-3)^2 < 0\), but a squared value is never negative, hence, there are also no solutions in Region 2.
04

Conclusion of the domain

All three regions we tested yielded no solutions. Hence, there are no real numbers x which satisfy the given expression. Therefore, the function does not have a domain in the real number system.

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!

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

Find the domain of the function \(\log _{2015}\left(1-\log _{7}\left(x^{2}-5 x+13\right)\right)\)

Let \(f(x)=x^{2}+x+\sin x-\cos x+\log _{e}(1+x)\) be defined on \([0,1]\). Find its even and odd extension in the interval \([-1,1]\)

Express the function \(f(x)=4^{\sin x}\) as a sum of an even and an odd function.

Find the period of \(f(x)=\frac{|\sin x+\cos x|}{|\sin x|-|\cos x|}\)

\(f(x)=\left(\frac{x}{e^{x}-1}+\frac{x}{2}+1\right)\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.