/*! 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 89 Find the domain of \(f\). \(f(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 89

Find the domain of \(f\). \(f(x)=\frac{x-5}{2 x+1}\)

Short Answer

Expert verified
The domain is \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\).

Step by step solution

01

Identify the Denominator

To find the domain of a rational function, identify the denominator because a function is undefined where its denominator is zero. Here, the denominator is \(2x + 1\).
02

Set the Denominator Equal to Zero

Set the denominator equal to zero and solve for \(x\) to find the value(s) that are excluded from the domain. \[2x + 1 = 0\]
03

Solve for \(x\)

Solve the equation \(2x + 1 = 0\) to find the excluded value: \(2x = -1\) \(x = -\frac{1}{2}\).
04

State the Domain

The domain of the function \(f(x)\) is all real numbers except the value found in Step 3. Therefore, the domain is: \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\).

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.

Denominator
In any rational function, the denominator plays a critical role because it determines where the function is undefined. By definition, a rational function is a fraction, and fractions are only undefined when the denominator equals zero. For the function provided, \(f(x)=\frac{x-5}{2x+1}\), the denominator is \(2x + 1\). Identifying this term is essential since solving for when it is zero helps determine which values of \(x\) make the function undefined. Always remember that the denominator must never be zero in a rational function.
Solving Equations
To find where the denominator is zero, you need to solve an equation. In this case, the equation is \(2x + 1 = 0\). Solving this equation involves isolating the variable \(x\). Here鈥檚 how you do it step-by-step:
  • First, you subtract 1 from both sides to get: \(2x = -1\)
  • Next, you divide both sides by 2 to isolate \(x\): \(x = -\frac{1}{2}\)

By following these steps, you determine the value of \(x\) that makes the denominator zero. This excluded value is crucial for defining the domain of the function.
Excluded Values
Excluded values are the x-values that make the denominator zero, rendering the function undefined. From the solved equation \(2x + 1 = 0\), we found that \(x = -\frac{1}{2}\). Hence, \(-\frac{1}{2}\) is the excluded value for the function \(f(x)=\frac{x-5}{2x+1}\). When stating the domain, you exclude this value to ensure the function remains defined.
Therefore, the domain of this function is all real numbers except \(-\frac{1}{2}\), written in interval notation as:
  • \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\)

This notation specifies that \(f(x)\) is defined for all values of \(x\) except \(-\frac{1}{2}\). Always exclude values that make the denominator zero.

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

Ultraviolet Index. At an ultraviolet, or UV, rating of \(4,\) those people who are less sensitive to the sun will burn in 75 min. Given that the number of minutes it takes to burn, \(t,\) varies inversely with the UV rating, \(u,\) how long will it take less sensitive people to burn when the UV rating is \(14 ?\)

The time \(T\) required to do a job varies inversely as the number of people \(P\) working. It takes 5 hr for 7 volunteers to pick up rubbish from 1 mi of roadway. How long would it take 10 volunteers to complete the job?

Calculate the slope of the line passing through the points \((a, f(a))\) and \((a+h, f(a+h))\) for the function \(f\) given by \(f(x)=3 x^{2} .\) Be sure your answer is simplified.

Find the variation constant and an equation of variation in which \(y\) varies inversely as \(x,\) and the following conditions exist. \(y=81\) when \(x=\frac{1}{9}\)

Simplify. Do not use negative exponents in the answer. $$ \frac{24 a^{-4} c^{-8}}{16 a^{5} c^{-7}} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.