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...
Related questions
Question
![### Producing a Rule for a Function from Its Graph
**Exercise:** Produce a rule for the function whose graph is shown. *(Hint: Consider using the basic functions and the transformations of their graphs.)*
**Graph Description:**
- The graph is plotted on a Cartesian coordinate system with the x-axis and y-axis marked.
- The graph consists of two distinct horizontal line segments.
**Segment Details:**
1. **Horizontal Line Segment 1:**
- This line segment is a straight horizontal line.
- It goes from \( x = -4 \) to \( x = 0 \) at \( y = 2 \).
- Ends with a solid dot at the point \( (0, 2) \).
2. **Horizontal Line Segment 2:**
- This line segment is also a straight horizontal line.
- It starts from \( x = 0 \) to \( x = 4 \) at \( y = 3 \).
- Starts with an open dot at the point \( (0, 3) \).
**Interpreting the Graph:**
- The function has two distinct constant values (2 and 3) based on the given x-intervals.
**Function Rule:**
To write a rule for the function, we use piecewise functions:
\[
f(x) =
\begin{cases}
2 & \text{if } -4 \leq x \leq 0 \\
3 & \text{if } 0 < x \leq 4
\end{cases}
\]
### Explanation:
- For \( x \) values between -4 and 0 inclusively, the function \( f(x) \) is constant at 2.
- For \( x \) values greater than 0 and up to 4, the function \( f(x) \) is constant at 3.
- The solid dot at \( (0, 2) \) indicates that the function value is 2 exactly at \( x = 0 \).
- The open dot at \( (0, 3) \) indicates the function value is not 3 at \( x = 0 \), making it an exclusive point.
### Graphical Analysis:
- The solid dot represents the inclusive boundary in piecewise function.
- The open dot represents the exclusive boundary in piecewise function.
### Application:
Understanding such graphs and functions helps in learning how piece](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8376e90-f370-4f87-9893-0898b7bef4cb%2F0a4a855d-64b6-4bfb-b55c-11b8f06f01ed%2Frnreqzk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Producing a Rule for a Function from Its Graph
**Exercise:** Produce a rule for the function whose graph is shown. *(Hint: Consider using the basic functions and the transformations of their graphs.)*
**Graph Description:**
- The graph is plotted on a Cartesian coordinate system with the x-axis and y-axis marked.
- The graph consists of two distinct horizontal line segments.
**Segment Details:**
1. **Horizontal Line Segment 1:**
- This line segment is a straight horizontal line.
- It goes from \( x = -4 \) to \( x = 0 \) at \( y = 2 \).
- Ends with a solid dot at the point \( (0, 2) \).
2. **Horizontal Line Segment 2:**
- This line segment is also a straight horizontal line.
- It starts from \( x = 0 \) to \( x = 4 \) at \( y = 3 \).
- Starts with an open dot at the point \( (0, 3) \).
**Interpreting the Graph:**
- The function has two distinct constant values (2 and 3) based on the given x-intervals.
**Function Rule:**
To write a rule for the function, we use piecewise functions:
\[
f(x) =
\begin{cases}
2 & \text{if } -4 \leq x \leq 0 \\
3 & \text{if } 0 < x \leq 4
\end{cases}
\]
### Explanation:
- For \( x \) values between -4 and 0 inclusively, the function \( f(x) \) is constant at 2.
- For \( x \) values greater than 0 and up to 4, the function \( f(x) \) is constant at 3.
- The solid dot at \( (0, 2) \) indicates that the function value is 2 exactly at \( x = 0 \).
- The open dot at \( (0, 3) \) indicates the function value is not 3 at \( x = 0 \), making it an exclusive point.
### Graphical Analysis:
- The solid dot represents the inclusive boundary in piecewise function.
- The open dot represents the exclusive boundary in piecewise function.
### Application:
Understanding such graphs and functions helps in learning how piece
![### Piecewise Function Definition Exercise
#### Graph Interpretation and Piecewise Function Input
This exercise provides students with a partially completed graph and an incomplete piecewise function definition to fill out.
#### Graph:
- The graph provided appears to be a Cartesian plane with a visible plot that the students need to carefully study. The graph extends from negative to positive values on both the x and y axes.
- Y-axis labels range from -4 to 2 in ascending order.
#### Piecewise Function:
Below the graph, there is a partially set-up piecewise function indicated by:
\[ f(x) = \begin{cases}
\boxed{} & \text{for } x \leq \boxed{} \\
\boxed{} & \text{for } x > \boxed{}
\end{cases} \]
*Note:* The boxed areas suggest where students need to enter specific values or expressions based on their interpretation of the graph given.
Students should observe the graph and determine the appropriate mathematical expressions and values to complete the piecewise function. The goal is to accurately describe the segments of the piecewise function according to the plotted graph.
#### Interactive Elements:
- There are two blue square input boxes for each piece of the piecewise function where students can type in their answers.
- The interface also includes a "Submit" (represented by 'X') and "Reset" button (indicated by a counter-clockwise arrow) that allows students to submit their answers or reset their inputs.
Students should use the graphical information provided to correctly determine the functions and their relevant domains.
They will enhance their understanding of how to interpret graphs and formally describe piecewise functions. This activity aims to improve comprehension in identifying intervals and the behavior of functions within those intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8376e90-f370-4f87-9893-0898b7bef4cb%2F0a4a855d-64b6-4bfb-b55c-11b8f06f01ed%2Fy72ht0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Function Definition Exercise
#### Graph Interpretation and Piecewise Function Input
This exercise provides students with a partially completed graph and an incomplete piecewise function definition to fill out.
#### Graph:
- The graph provided appears to be a Cartesian plane with a visible plot that the students need to carefully study. The graph extends from negative to positive values on both the x and y axes.
- Y-axis labels range from -4 to 2 in ascending order.
#### Piecewise Function:
Below the graph, there is a partially set-up piecewise function indicated by:
\[ f(x) = \begin{cases}
\boxed{} & \text{for } x \leq \boxed{} \\
\boxed{} & \text{for } x > \boxed{}
\end{cases} \]
*Note:* The boxed areas suggest where students need to enter specific values or expressions based on their interpretation of the graph given.
Students should observe the graph and determine the appropriate mathematical expressions and values to complete the piecewise function. The goal is to accurately describe the segments of the piecewise function according to the plotted graph.
#### Interactive Elements:
- There are two blue square input boxes for each piece of the piecewise function where students can type in their answers.
- The interface also includes a "Submit" (represented by 'X') and "Reset" button (indicated by a counter-clockwise arrow) that allows students to submit their answers or reset their inputs.
Students should use the graphical information provided to correctly determine the functions and their relevant domains.
They will enhance their understanding of how to interpret graphs and formally describe piecewise functions. This activity aims to improve comprehension in identifying intervals and the behavior of functions within those intervals.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
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
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
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
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning