/*! 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 26 (a) state the domain of the func... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 26

(a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. $$f(x)=-\frac{1}{(x-2)^{2}}$$

Short Answer

Expert verified
The domain of the function is all real numbers except 2. There are no x-intercepts and the y-intercept is at -1/4. The function has a vertical asymptote at \(x = 2\) and a horizontal asymptote at \(y = 0\). The graph will appear in the third and second quadrants when \(x<2\), and in the first and fourth quadrants when \(x>2\).

Step by step solution

01

Finding the Domain

The domain of a function consists of all the real values of \(x\) that will make the function \(f(x)\) real. For a fraction to be real, the denominator must not be zero. So to find the domain, set the denominator equal to zero and solve for \(x: (x-2)^2=0\). From this, \(x=2\). Thus, the domain of \(f(x)\) is all real numbers except 2, which can be written as \(-\infty < x < 2\) and \(2 < x < \infty\).
02

Identifying the Intercepts

The x-intercept of the function is found by setting \(f(x) = 0\). However, a fraction is only zero when the numerator is zero, but in this case, the numerator is a constant (-1) and not equal to zero, which means there are no x-intercepts. The y-intercept is found by setting \(x = 0\) in the function. But \(f(0) = -\frac{1}{(0-2)^2} = -\frac{1}{4}\). Thus, y-intercept is -1/4.
03

Finding Asymptotes

Vertical asymptotes occur at the values of \(x\) that make the function undefined. For this function, \(x = 2\) is a vertical asymptote. Horizontal asymptotes are determined by looking at the degrees of the numerator and denominator. Here, the degree of the numerator is less than the denominator, hence the horizontal asymptote is \(y = 0\).
04

Sketching the Graph

Using the previously calculated information (domain, intercepts, and asymptotes), you can sketch the graph of the function. Remember that the function approaches but never reaches the asymptotes. Also plot the y-intercept at -1/4. For \(x<2\), the graph will be in the third and second quadrants, and for \(x>2\), it will be in the first and fourth quadrants.

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 Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=5 x^{5}-10 x$$

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$

The mean salaries \(S\) (in thousands of dollars) of public school classroom teachers in the United States from 2000 through 2011 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, \(S\) } \\\\\hline 2000 & 42.2 \\\2001 & 43.7 \\\2002 & 43.8 \\\2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\2008 & 51.3 \\\2009 & 52.9 \\\2010 & 54.4 \\\2011 & 54.2 \\\\\hline\end{array}$$ A model that approximates these data is given by $$S=\frac{42.16-0.236 t}{1-0.026 t}, \quad 0 \leq t \leq 11$$ where \(t\) represents the year, with \(t=0\) corresponding to 2000. (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) Use the model to predict when the salary for classroom teachers will exceed \(\$ 60,000\). (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

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.