/*! 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 90 Find the domain of \(f\). \(f(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 90

Find the domain of \(f\). \(f(x)=\frac{3 x}{x^{2}+1}\)

Short Answer

Expert verified
The domain of \(f(x) = \frac{3x}{x^2 + 1}\) is all real numbers.

Step by step solution

01

Understand the function

The given function is \( f(x) = \frac{3x}{x^2 + 1} \). To find the domain, determine the values of \(x\) for which the function is defined.
02

Identify potential restrictions

Potential restrictions in the domain of a function often arise from divisions by zero or roots of negative numbers. Here, the expression \(x^2 + 1\) is in the denominator.
03

Analyze the denominator

Determine if \(x^2 + 1\) can ever be zero. Solve the equation \(x^2 + 1 = 0\).
04

Solve for zero denominator

Solving \(x^2 + 1 = 0\), we get \(x^2 = -1\). Since \(x^2\) for real numbers is always non-negative, there are no real solutions to this equation.
05

Conclude the domain

Since \(x^2 + 1\) is never zero for any real number \(x\), the function is defined for all real numbers.

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.

headline of the respective core concept
Learning about numerical expressions is crucial when analyzing functions like \( f(x) = \frac{3x}{x^2 + 1} \). Numerical expressions contain numbers and mathematical operations but no variables. For example, when simplifying \( x^2 + 1 \), you're not changing the expression, just understanding its structure. Another example is the numerator \( 3x \), which is a simpler expression. Understanding these allows you to break down complex functions easier. When solving problems step-by-step, numerical expressions help by focusing on specific parts of the problem. This ensures clarity and accuracy as you work through each step.
headline of the respective core concept
Denominators play a critical role in functions involving fractions, like \( f(x) = \frac{3x}{x^2 + 1} \). The denominator is the part of the fraction below the line, dictating how many parts the numerator is divided into. A key point is ensuring the denominator never equals zero, which would make the function undefined. In this example, we examine \( x^2 + 1 \). Since squaring any real number results in a non-negative number, adding 1 makes \( x^2 + 1 \) always positive. Hence, there is no risk of division by zero, so the function is defined for all real numbers. This understanding is vital for identifying the domain.
headline of the respective core concept
Real numbers encompass all numbers you can think of on a number line, including both rational and irrational numbers. When determining the domain of a function, you check which real numbers you can input without breaking any mathematical rules. For the function \( f(x) = \frac{3x}{x^2 + 1} \), we analyze whether the denominator \( x^2 + 1 \) could equal zero, which it can't for any real number. This absence of value causing a division by zero means that the function accepts every real number as an input, making its domain all real numbers. Understanding real numbers helps you confidently determine the range of inputs for any function.

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 variation constant and an equation of variation in which \(y\) varies inversely as \(x,\) and the following conditions exist. \(y=40\) when \(x=8\)

Simplify. $$ \frac{x^{5}-x^{3}+x^{2}-1-\left(x^{3}-1\right)(x+1)^{2}}{\left(x^{2}-1\right)^{2}} $$

Find and simplify $$ \frac{f(x+h)-f(x)}{h} $$ for each rational function \(f\) $$ f(x)=\frac{3}{x} $$

The wavelength \(W\) of a radio wave varies inversely as its frequency \(F\) A wave with a frequency of 1200 kilohertz has a length of 300 meters. What is the length of a wave with a frequency of 800 kilohertz?

Simplify. Do not use negative exponents in the answer. $$ -3^{-2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.