/*! 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 88 Find the domain of the function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 88

Find the domain of the function given by each equation. $$ f(x)=\frac{5}{2 x+1} $$

Short Answer

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

Step by step solution

01

- Understand the function

The function given is a rational function: \[ f(x) = \frac{5}{2x + 1} \] The domain of a rational function is all real numbers except where the denominator is zero.
02

- Set the denominator not equal to zero

To find the values of \(x\) for which the function is undefined, set the denominator equal to zero and solve. \[ 2x + 1 eq 0 \]
03

- Solve for x

Solve the equation obtained in Step 2.\[ 2x + 1 eq 0 \]Subtract 1 from both sides:\[ 2x eq -1 \]Divide by 2:\[ x eq -\frac{1}{2} \]
04

- Write the domain

Since the function is not defined at \( x = -\frac{1}{2} \), the domain is all real numbers except \( -\frac{1}{2} \).
05

- Express the domain

Express the domain in interval notation, excluding \( x = -\frac{1}{2} \):\[ (-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty) \]

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.

Rational Functions
A rational function is any function that can be represented as the quotient of two polynomials. In other words, a rational function takes the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Rational functions are important due to their wide range of applications, like solving real-world problems in physics and economics. But they come with a tricky aspect - their domain. Specifically, rational functions can have restrictions on their domain, which we'll discuss next.
Domain Restrictions
The domain of a function is the set of all possible input values (or \( x \) values) that the function can accept without causing any issues, like division by zero. In rational functions, domain restrictions occur when the denominator equals zero, causing the function to be undefined. To find the domain of a rational function like \( f(x) = \frac{5}{2x + 1} \), you need to set the denominator not equal to zero: \( 2x + 1 e 0 \). Solving \( 2x + 1 e 0 \) for \( x \) gives \( x e -\frac{1}{2} \). Thus, the function is undefined at \( x = -\frac{1}{2} \) and we exclude this value from the domain.
Interval Notation
Interval notation is a way of writing the domain of a function that clearly shows the range of values allowed for the input \( x \). When using interval notation, we express the domain by listing intervals separated by the union symbol \( \bigcup \). For our function \( f(x) = \frac{5}{2x + 1} \), the domain in interval notation would be all real numbers except \( x = -\frac{1}{2} \): \( (-\text\infty, -\frac{1}{2}) \bigcup (-\frac{1}{2}, \text\infty) \). This notation is helpful because it is clear and concise, allowing us to communicate domain restrictions efficiently.

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

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 8 x+40=0 $$

Combine like terms. [ 1.3] $$-6 x-x-2 x$$

Graph equation after plotting at least 10 points. \(y=\sqrt{x}+1 ;\) use \(x\) -values from 0 to 10

Use a commutative law to write an expression equivalent to $7+t

Solve. If appropriate, classify the equation as either \(a\) contradiction or an identity. $$ 9-6(x-7)=12 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure 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.