/*! 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 25 Find the vertical asymptotes, if... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 25

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x+4)}$$

Short Answer

Expert verified
The function \(h(x)=\frac{x}{x(x+4)}\) has a vertical asymptote at \(x = -4\) and no holes.

Step by step solution

01

Simplifying the function

First, simplify the function by cancelling out any common factors in the numerator and the denominator. The function \(h(x)=\frac{x}{x(x+4)}\) simplifies to \(h(x)=\frac{1}{x+4}\) because the \(x\)s cancel out.
02

Solving for the vertical asymptote

A function will have vertical asymptotes where the denominator is equal to zero, provided the factors causing this don鈥檛 cancel out. Solve the equation \(x+4 = 0\) for \(x\). The solution \(x = -4\) is a vertical asymptote.
03

Finding the hole

A function will have a hole where the denominator is equal to zero, but the factors causing this do cancel out. As there are no factors that resulted in division by zero left, there are no holes in the graph of this function.

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 everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\)

What is a rational function?

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{x+2}{x-3} \leq 2 $$

You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.