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
![**Graph and Label the Function**
Given the piecewise function \( g(x) \):
\[
g(x) =
\begin{cases}
3x + 4, & x < -2 \\
5, & x = -2 \\
-x + 1, & x > -2
\end{cases}
\]
### Explanation:
- **For \( x < -2 \):** The function is defined as \( g(x) = 3x + 4 \). This is a linear equation with a slope of 3 and a y-intercept of 4.
- **At \( x = -2 \):** The function takes on a constant value, \( g(x) = 5 \). This represents a single point on the graph at (-2, 5).
- **For \( x > -2 \):** The function is \( g(x) = -x + 1 \). This is also a linear equation, but with a slope of -1 and a y-intercept of 1.
When graphing, ensure to:
1. Plot the line \( 3x + 4 \) for all x-values less than -2.
2. Mark the point (-2, 5) on the graph, representing the value of the function at \( x = -2 \).
3. Plot the line \(-x + 1\) for all x-values greater than -2.
Use open and closed circles as necessary to indicate whether endpoints are included in each piece of the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49520758-34b8-4feb-b56e-1bcd3eb1e5da%2Fa77b357f-02c4-4019-8baf-f4195d91c507%2Ffy9tq6_processed.png&w=3840&q=75)
Transcribed Image Text:**Graph and Label the Function**
Given the piecewise function \( g(x) \):
\[
g(x) =
\begin{cases}
3x + 4, & x < -2 \\
5, & x = -2 \\
-x + 1, & x > -2
\end{cases}
\]
### Explanation:
- **For \( x < -2 \):** The function is defined as \( g(x) = 3x + 4 \). This is a linear equation with a slope of 3 and a y-intercept of 4.
- **At \( x = -2 \):** The function takes on a constant value, \( g(x) = 5 \). This represents a single point on the graph at (-2, 5).
- **For \( x > -2 \):** The function is \( g(x) = -x + 1 \). This is also a linear equation, but with a slope of -1 and a y-intercept of 1.
When graphing, ensure to:
1. Plot the line \( 3x + 4 \) for all x-values less than -2.
2. Mark the point (-2, 5) on the graph, representing the value of the function at \( x = -2 \).
3. Plot the line \(-x + 1\) for all x-values greater than -2.
Use open and closed circles as necessary to indicate whether endpoints are included in each piece of the function.
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