Homework Help is Here – Start Your Trial Now!
Math
Calculus
Which of the following is the graph of the piecewise function f (x) - 7 f(x) -2-1 2 1 √-x-1 if XS-1 X x²-x-2 log₂ (x-1) if if -1

Which of the following is the graph of the piecewise function f (x) - 7 f(x) -2-1 2 1 √-x-1 if XS-1 X x²-x-2 log₂ (x-1) if if -1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
### Graphing Piecewise Functions **Question:** Which of the following is the graph of the piecewise function \( f(x) \)? \[ f(x) = \begin{cases} \sqrt{-x-1} & \text{if } x \leq -1 \\ x & \text{if } -1 < x < 2 \\ x^2 - x - 2 & \text{if } x \geq 2 \\ \log_{2}(x-1) & \text{if } x \geq 2 \end{cases} \] **Explanation of Diagram:** The image contains two graphs, each representing a function on a set of Cartesian coordinates. 1. **Graph 1:** - The graph consists of four distinct parts corresponding to the four pieces of the piecewise function. - **For \( x \leq -1 \):** The graph portrays the function \( \sqrt{-x-1} \), showing a curve starting from \( (-1, 0) \) and extending to the left and downward. - **For \( -1 < x < 2 \):** The graph depicts the linear function \( f(x) = x \), forming a straight line passing through the points around \( x = 0, y = 0 \), and up towards \( x = 2 \). - **For \( x \geq 2 \):** The graph extends from \( x = 2 \) at point (2,0), plotting the function \( \log_{2}(x-1) \). This forms a curve extending to the right, accordingly matching \( \log_{2}\) characteristics. 2. **Graph 2:** - The graph consists of a different plotting depicting the piecewise function differently. - **For \( x \leq -1 \):** It also graphs \( \sqrt{-x-1} \) from \( (-1,0) \) extending left and downwards. - **For \( -1 < x < 2 \):** This part is a linear straight line extending from around zero and near \( x= 1 \). - **For \( x \geq 2 \):** It follows a curve representing \( x^2 - x - 2 \) from small positive values around 2 extending beyond
Transcribed Image Text:### Graphing Piecewise Functions **Question:** Which of the following is the graph of the piecewise function \( f(x) \)? \[ f(x) = \begin{cases} \sqrt{-x-1} & \text{if } x \leq -1 \\ x & \text{if } -1 < x < 2 \\ x^2 - x - 2 & \text{if } x \geq 2 \\ \log_{2}(x-1) & \text{if } x \geq 2 \end{cases} \] **Explanation of Diagram:** The image contains two graphs, each representing a function on a set of Cartesian coordinates. 1. **Graph 1:** - The graph consists of four distinct parts corresponding to the four pieces of the piecewise function. - **For \( x \leq -1 \):** The graph portrays the function \( \sqrt{-x-1} \), showing a curve starting from \( (-1, 0) \) and extending to the left and downward. - **For \( -1 < x < 2 \):** The graph depicts the linear function \( f(x) = x \), forming a straight line passing through the points around \( x = 0, y = 0 \), and up towards \( x = 2 \). - **For \( x \geq 2 \):** The graph extends from \( x = 2 \) at point (2,0), plotting the function \( \log_{2}(x-1) \). This forms a curve extending to the right, accordingly matching \( \log_{2}\) characteristics. 2. **Graph 2:** - The graph consists of a different plotting depicting the piecewise function differently. - **For \( x \leq -1 \):** It also graphs \( \sqrt{-x-1} \) from \( (-1,0) \) extending left and downwards. - **For \( -1 < x < 2 \):** This part is a linear straight line extending from around zero and near \( x= 1 \). - **For \( x \geq 2 \):** It follows a curve representing \( x^2 - x - 2 \) from small positive values around 2 extending beyond
**Understanding Graphs of Rational Functions** In this tutorial, we will examine two graphs of a function \( f(x) \), which is a rational function. These graphs will help in understanding how the function behaves around its asymptotes. ### Graph Analysis #### First Graph The first graph shows the function \( f(x) \) with the following characteristics: - On the x-axis ranging from -8 to 8, and - On the y-axis ranging from -8 to 8. **Key Features:** 1. **Vertical Asymptote:** There is a vertical asymptote at \( x = 2 \). This is evident because the function tends toward positive or negative infinity as \( x \) approaches 2 from either side. 2. **Horizontal Asymptote:** There is a horizontal asymptote at \( y = 2 \). This indicates that as \( x \) grows larger in both the positive and negative directions, \( f(x) \) approaches 2. 3. **Behavior Near Asymptotes:** - As \( x \to 2^- \), \( f(x) \to -\infty \). - As \( x \to 2^+ \), \( f(x) \to -\infty \). - As \( x \to \infty \ or \ x \to -\infty \), \( f(x) \to 2 \). #### Second Graph The second graph is almost identical to the first and shows the function \( f(x) \) with the same x and y axis ranges. **Key Features:** 1. **Vertical Asymptote:** Vertical asymptote is present at \( x = 1 \). 2. **Horizontal Asymptote:** There is a horizontal asymptote at \( y = -3 \). 3. **Behavior Near Asymptotes:** - As \( x \to 1^- \), \( f(x) \to \infty \). - As \( x \to 1^+ \), \( f(x) \to -\infty \). - As \( x \to \infty \ or \ x \to -\infty \), \( f(x) \to -3 \). From these descriptions and analyses, we have identified the locations of the vertical and horizontal asymptotes using the graphs provided. Understanding
Transcribed Image Text:**Understanding Graphs of Rational Functions** In this tutorial, we will examine two graphs of a function \( f(x) \), which is a rational function. These graphs will help in understanding how the function behaves around its asymptotes. ### Graph Analysis #### First Graph The first graph shows the function \( f(x) \) with the following characteristics: - On the x-axis ranging from -8 to 8, and - On the y-axis ranging from -8 to 8. **Key Features:** 1. **Vertical Asymptote:** There is a vertical asymptote at \( x = 2 \). This is evident because the function tends toward positive or negative infinity as \( x \) approaches 2 from either side. 2. **Horizontal Asymptote:** There is a horizontal asymptote at \( y = 2 \). This indicates that as \( x \) grows larger in both the positive and negative directions, \( f(x) \) approaches 2. 3. **Behavior Near Asymptotes:** - As \( x \to 2^- \), \( f(x) \to -\infty \). - As \( x \to 2^+ \), \( f(x) \to -\infty \). - As \( x \to \infty \ or \ x \to -\infty \), \( f(x) \to 2 \). #### Second Graph The second graph is almost identical to the first and shows the function \( f(x) \) with the same x and y axis ranges. **Key Features:** 1. **Vertical Asymptote:** Vertical asymptote is present at \( x = 1 \). 2. **Horizontal Asymptote:** There is a horizontal asymptote at \( y = -3 \). 3. **Behavior Near Asymptotes:** - As \( x \to 1^- \), \( f(x) \to \infty \). - As \( x \to 1^+ \), \( f(x) \to -\infty \). - As \( x \to \infty \ or \ x \to -\infty \), \( f(x) \to -3 \). From these descriptions and analyses, we have identified the locations of the vertical and horizontal asymptotes using the graphs provided. Understanding
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS