/*! 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 8 Find the domain of the function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 8

Find the domain of the function and discuss the behavior of \(f\) near any excluded \(x\) -values. $$f(x)=\frac{2 x}{x^{2}-4}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\frac{2 x}{x^{2}-4}\) is \(-鈭�<x<-2\), \(-2<x<2\), \(2<x<鈭瀄). At \(x=-2\) and \(x=2\), the function tends to respective positive and negative infinities, indicating asymptotic behavior at these points.

Step by step solution

01

Determine the Domain

To find the domain of \(f(x)=\frac{2 x}{x^{2}-4}\), set the denominator equal to zero and solve the equation. \(x^{2}-4=0\). This will help to find the values of \(x\) that are undefined.
02

Solve the Equation

The equation \(x^{2}-4=0\) can be factored as \((x-2)(x+2)=0\). Solving for \(x\), we get \(x=2\), \(x=-2\). These are the points where the function is undefined.
03

Write the Domain

The domain of a function is made up of all real numbers except the points where the function is undefined. For this function, \(x\) is defined for all real values except \(2\) and \(-2\). Hence, the domain is \(-鈭�<x<-2\) or \(-2<x<2\) or \(2<x<鈭瀄)
04

Analyze the behavior of f near the excluded x-values

To determine the behavior of the function near \(x=2\) and \(x=-2\), you can consider values of \(x\) that approach these points from both sides. Mathematically, this consideration is done by evaluating \(\lim_{{x 鈫� 2^-}}f(x)\), \(\lim_{{x 鈫� 2^+}}f(x)\), \(\lim_{{x 鈫� -2^-}}f(x)\), \(\lim_{{x 鈫� -2^+}}f(x)\). This will help to determine whether the function is approaching positive or negative infinity at the singularity points. If the limit from both the negative and positive sides at a singularity is the same (either both go to positive infinity or negative infinity), then the function shows an asymptotic behavior towards that infinity at the singularity.

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

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=2 x^{3}-3 x^{2}-3$$

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$

Prove that the complex conjugate of the sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the sum of their complex conjugates.

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{3}-x^{2}+x+39$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(s)=2 s^{3}-5 s^{2}+12 s-5$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.