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
![The text on the screen is as follows:
---
"Suppose that the function \( g \) is defined, for all real numbers, as follows.
\[
g(x) =
\begin{cases}
-1 & \text{if } x \neq 0 \\
-2 & \text{if } x = 0
\end{cases}
\]
Graph the function \( g \)."
---
Below the text, there is a coordinate plane with the x-axis and y-axis. The graph is expected to represent the function \( g(x) \) as defined above.
Explanation of the graph:
- For all values of \( x \) not equal to 0, \( g(x) = -1 \). This means the graph will have a horizontal line at \( y = -1 \) for \( x \neq 0 \).
- At \( x = 0 \), \( g(x) = -2 \). Therefore, there will be a distinct point at \( (0, -2) \).
- The graph will show a continuous line at \( y = -1 \) with an isolated point at \( (0, -2) \).
To visually distinguish \( x = 0 \), you might see a filled or open circle on the graph to represent the change in value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34aa3c00-4e68-4586-905d-c19fda080a0f%2F760ef238-74f6-4548-b0de-09ef629edfd6%2Fafkg22_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The text on the screen is as follows:
---
"Suppose that the function \( g \) is defined, for all real numbers, as follows.
\[
g(x) =
\begin{cases}
-1 & \text{if } x \neq 0 \\
-2 & \text{if } x = 0
\end{cases}
\]
Graph the function \( g \)."
---
Below the text, there is a coordinate plane with the x-axis and y-axis. The graph is expected to represent the function \( g(x) \) as defined above.
Explanation of the graph:
- For all values of \( x \) not equal to 0, \( g(x) = -1 \). This means the graph will have a horizontal line at \( y = -1 \) for \( x \neq 0 \).
- At \( x = 0 \), \( g(x) = -2 \). Therefore, there will be a distinct point at \( (0, -2) \).
- The graph will show a continuous line at \( y = -1 \) with an isolated point at \( (0, -2) \).
To visually distinguish \( x = 0 \), you might see a filled or open circle on the graph to represent the change in value.
![The problem statement is as follows:
Suppose that the function \( f \) is defined for all real numbers as follows:
\[
f(x) =
\begin{cases}
1 - x^2 & \text{if} \ x < 1 \\
-2x - 3 & \text{if} \ x \geq 1
\end{cases}
\]
You are asked to graph the function \( f \). Then determine whether or not the function is continuous.
### Explanation of the Graph:
The graph on the screen appears to consist of two parts:
1. **Quadratic Function**: \( f(x) = 1 - x^2 \) for \( x < 1 \)
- This portion of the graph is part of a downward opening parabola. It will have its vertex at the point (0, 1) and crosses the x-axis at x = ±1. Only the section for \( x < 1 \) is relevant here.
2. **Linear Function**: \( f(x) = -2x - 3 \) for \( x \geq 1 \)
- This part of the graph is a straight line with a negative slope. It passes through the point (1, -5) and continues downward.
You should evaluate the function at \( x = 1 \) to determine continuity:
- From the left (using \( 1 - x^2 \)), \( f(1^{-}) = 1 - 1^2 = 0 \)
- From the right (using \(-2x - 3\)), \( f(1^{+}) = -2(1) - 3 = -5 \)
Since the limits from the left and right do not match, the function is discontinuous at \( x = 1 \).
The screen also includes graphing tools, such as line, curve, and point selectors, indicating that you can use these to construct the graph visually.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34aa3c00-4e68-4586-905d-c19fda080a0f%2F760ef238-74f6-4548-b0de-09ef629edfd6%2F6megz2n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem statement is as follows:
Suppose that the function \( f \) is defined for all real numbers as follows:
\[
f(x) =
\begin{cases}
1 - x^2 & \text{if} \ x < 1 \\
-2x - 3 & \text{if} \ x \geq 1
\end{cases}
\]
You are asked to graph the function \( f \). Then determine whether or not the function is continuous.
### Explanation of the Graph:
The graph on the screen appears to consist of two parts:
1. **Quadratic Function**: \( f(x) = 1 - x^2 \) for \( x < 1 \)
- This portion of the graph is part of a downward opening parabola. It will have its vertex at the point (0, 1) and crosses the x-axis at x = ±1. Only the section for \( x < 1 \) is relevant here.
2. **Linear Function**: \( f(x) = -2x - 3 \) for \( x \geq 1 \)
- This part of the graph is a straight line with a negative slope. It passes through the point (1, -5) and continues downward.
You should evaluate the function at \( x = 1 \) to determine continuity:
- From the left (using \( 1 - x^2 \)), \( f(1^{-}) = 1 - 1^2 = 0 \)
- From the right (using \(-2x - 3\)), \( f(1^{+}) = -2(1) - 3 = -5 \)
Since the limits from the left and right do not match, the function is discontinuous at \( x = 1 \).
The screen also includes graphing tools, such as line, curve, and point selectors, indicating that you can use these to construct the graph visually.
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