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...
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![### Step-by-step Graphing of Piecewise Function
**Function Definition:**
The piecewise function \( n(x) \) is defined as follows:
\[
n(x) =
\begin{cases}
-4 & \text{for } -3 < x < -1 \\
x & \text{for } -1 \le x < 1 \\
-x^2 + 1 & \text{for } x \ge 1
\end{cases}
\]
**Graph Description:**
The graphing area is divided into three regions according to the piecewise definition of the function \( n(x) \):
1. **First Region: \( -3 < x < -1 \)**
- In this interval, the function \( n(x) \) is constant and equals -4.
- The corresponding graph segment will be a horizontal line at \( y = -4 \).
2. **Second Region: \( -1 \le x < 1 \)**
- Here, the function \( n(x) \) follows the line described by \( y = x \).
- This segment forms a straight line passing through the origin (0,0) with a slope of 1.
- This line only exists from \( x = -1 \) to \( x = 1 \).
3. **Third Region: \( x \ge 1 \)**
- In this interval, the function is a downward-facing parabola described by \( y = -x^2 + 1 \).
- The parabola opens downwards and has its vertex at (1, 0), reaching a maximum point at \( y = 1 \) when \( x = 1 \).
**Visual Representation:**
- The provided image features a square coordinate grid extending from \( x = -4 \) to \( x = 4 \) on the x-axis and from \( y = -4 \) to \( y = 4 \) on the y-axis.
- A vertical bar on the graphing tools interface provides options for drawing and manipulating graphical elements such as lines, curves, and shapes.
By plotting each piece of the function on the respective intervals, we will achieve a visual representation that effectively communicates the behavior of the piecewise function \( n(x) \).
The process of graphing this piecewise function involves:
1. Drawing a horizontal line at \( y = -4 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64e8698f-bf7c-4e1b-8c7e-b7a3a74b0e52%2Fed226cd2-62d3-471d-b4a7-c16c9e679b8e%2Fv1h05pr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Step-by-step Graphing of Piecewise Function
**Function Definition:**
The piecewise function \( n(x) \) is defined as follows:
\[
n(x) =
\begin{cases}
-4 & \text{for } -3 < x < -1 \\
x & \text{for } -1 \le x < 1 \\
-x^2 + 1 & \text{for } x \ge 1
\end{cases}
\]
**Graph Description:**
The graphing area is divided into three regions according to the piecewise definition of the function \( n(x) \):
1. **First Region: \( -3 < x < -1 \)**
- In this interval, the function \( n(x) \) is constant and equals -4.
- The corresponding graph segment will be a horizontal line at \( y = -4 \).
2. **Second Region: \( -1 \le x < 1 \)**
- Here, the function \( n(x) \) follows the line described by \( y = x \).
- This segment forms a straight line passing through the origin (0,0) with a slope of 1.
- This line only exists from \( x = -1 \) to \( x = 1 \).
3. **Third Region: \( x \ge 1 \)**
- In this interval, the function is a downward-facing parabola described by \( y = -x^2 + 1 \).
- The parabola opens downwards and has its vertex at (1, 0), reaching a maximum point at \( y = 1 \) when \( x = 1 \).
**Visual Representation:**
- The provided image features a square coordinate grid extending from \( x = -4 \) to \( x = 4 \) on the x-axis and from \( y = -4 \) to \( y = 4 \) on the y-axis.
- A vertical bar on the graphing tools interface provides options for drawing and manipulating graphical elements such as lines, curves, and shapes.
By plotting each piece of the function on the respective intervals, we will achieve a visual representation that effectively communicates the behavior of the piecewise function \( n(x) \).
The process of graphing this piecewise function involves:
1. Drawing a horizontal line at \( y = -4 \
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