/*! 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 the function ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 90

Find the domain of the function given by each equation. $$ h(x)=\frac{3 x}{x+7} $$

Short Answer

Expert verified
The domain of \( h(x) = \frac{3x}{x+7} \) is \( (-\infty, -7) \cup (-7, +\infty) \).

Step by step solution

01

Title - Understand the Function

Identify the given function. We have the function \( h(x) = \frac{3x}{x+7} \).
02

Title - Identify Restrictions

Look for values that make the denominator zero because division by zero is not defined. Set the denominator equal to zero and solve for \( x \). So, solve \( x + 7 = 0 \).
03

Title - Solve the Equation

Solving for \( x \), we get \( x = -7 \). This means the function is not defined at \( x = -7 \).
04

Title - Write the Domain

The domain of \( h(x) \) includes all real numbers except \( x = -7 \). This can be written in interval notation as \( (-\infty, -7) \cup (-7, +\infty) \).

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
Rational functions are fractions where both the numerator and the denominator are polynomials. The general form is \[ f(x) = \frac{P(x)}{Q(x)} \], where \[ P(x) \] and \[ Q(x) \] are polynomial functions. For example, in the function \[ h(x) = \frac{3x}{x+7} \], the numerator \[ 3x \] and the denominator \[ x+7 \] are both polynomials. Rational functions can have interesting behaviors, like vertical asymptotes and holes, depending on the values that make the denominator zero. These particular values are essential when understanding the domain of the function.
Domain Restrictions
Domain restrictions occur when certain values make the function undefined. For rational functions, this usually happens when the denominator is zero because division by zero is undefined in mathematics. To find these restrictions, you set the denominator equal to zero and solve for \[ x \]. Let's look at the function \[ h(x) = \frac{3x}{x+7} \]. To find where this function is undefined, solve the equation \[ x + 7 = 0 \]. Solving for \[ x \] we get \[ x = -7 \]. This is the value where the function cannot exist. Hence, \[ x = -7 \] is a domain restriction, and it must be excluded from the domain of \[ h(x) \].
Interval Notation
Interval notation is a way of writing subsets of the real number line. This system is often used to express the domain of a function. With interval notation, we use parentheses \[ ( ) \] to exclude endpoints and brackets \[ [ ] \] to include them. For the function \[ h(x) = \frac{3x}{x+7} \], we need to exclude \[ x = -7 \] since the function is not defined there. So, the domain includes all real numbers except \[ -7 \]. Using interval notation, we write this as \[ (-\infty, -7) \cup (-7, +\infty) \]. The union symbol \[ \cup \] indicates the combination of these two intervals, ensuring \[ -7 \] is not included in the domain.

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 given by each equation. $$ f(x)=\frac{4 x-3}{5} $$

For Exercises 103鈥�107, assume that a linear equation models each situation. Temperature Conversion. Water freezes at \(32^{\circ}\) Fahrenheit and at \(0^{\circ}\) Celsius. Water boils at \(212^{\circ} \mathrm{F}\) and at \(100^{\circ} \mathrm{C}\). What Celsius temperature corresponds to a room temperature of \(70^{\circ} \mathrm{F} ?\)

Find the indicated function values. $$ f(x)=\left\\{\begin{array}{ll}{x^{2}-10,} & {\text { if } x<-10} \\ {x^{2},} & {\text { if }-10 \leq x \leq 10} \\ {x^{2}+10,} & {\text { if } x>10}\end{array}\right. $$ a) \(f(-10) \quad\) b) \(f(10)\) c) \(f(11)\)

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

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ 4 x+1=-x+11 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.