8. Which of the following is the equation of the piecewise linear function shown below? y x+4 x<2 (1) f(x)=- 3x+5 x22 1. ーx+4 x<2 (2) f(x)={2 3x-1 x> 2 1 x+5 x<2 4 (3) f(x)=: 3x-3 x22 -2x+4 x<2 (4) f(x)= 4x+1 x22
8. Which of the following is the equation of the piecewise linear function shown below? y x+4 x<2 (1) f(x)=- 3x+5 x22 1. ーx+4 x<2 (2) f(x)={2 3x-1 x> 2 1 x+5 x<2 4 (3) f(x)=: 3x-3 x22 -2x+4 x<2 (4) f(x)= 4x+1 x22
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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Transcribed Image Text:**Problem 8: Identify the Equation of the Piecewise Linear Function**
- The graph consists of two line segments, displayed on a standard Cartesian coordinate system. The first segment is linear with a generally increasing slope before reaching x = 2. The second segment continues from x = 2 with a different slope.
**Choose the correct equation from the options below:**
1) \( f(x) = \begin{cases}
x + 4 & x < 2 \\
3x + 5 & x \geq 2
\end{cases} \)
2) \( f(x) = \begin{cases}
\frac{1}{2}x + 4 & x < 2 \\
3x - 1 & x \geq 2
\end{cases} \)
3) \( f(x) = \begin{cases}
-\frac{1}{4}x + 5 & x < 2 \\
3x - 3 & x \geq 2
\end{cases} \)
4) \( f(x) = \begin{cases}
-2x + 4 & x < 2 \\
4x + 1 & x \geq 2
\end{cases} \)
**Diagram Explanation:**
The graph shows a piecewise linear function with distinct behaviors in different segments of the x-axis. Each segment is defined by its own linear equation, indicating a change in slope when x equals 2.
---
**Problem 9: Symmetry of Functions and Their Inverses**
- The graphs of a function and its inverse are symmetric across which line?
Select the correct line of symmetry:
1) The x-axis
2) The y-axis
3) The line \( y = x \)
4) The line \( y = -x \)
**Note:**
The diagram reinforces the standard behavior of functions and their inverses, highlighting the geometrical reflection that occurs.
![### Educational Content: Analyzing Linear Functions and Their Inverses
#### Problem 10: Linear Graph Analysis
**Given Instructions**:
- Analyze the linear graph provided and answer the following questions.
**(a) Equation of the Line**:
- Write the equation of the line in \( y = mx + b \) form.
**Solution**:
1. By observing the graph, determine the slope (\(m\)) and y-intercept (\(b\)).
2. From the calculations shown, the slope \(m\) is given by:
\[
m = \frac{6 - 0}{0 + 6} = \frac{1}{2}
\]
3. The equation, considering the y-intercept (-6), is:
\[
y = \frac{1}{2}x - 6
\]
**(b) Inverse Graph**:
- Create a graph of this linear function's inverse on the same set of graph paper.
**Explanation**:
- To graph the inverse, \( y = \frac{1}{2}x - 6 \) should be swapped to \( x = \frac{1}{2}y - 6 \) and solved for \( y \).
**(c) Equation of the Inverse**:
- Determine the equation of the inverse.
**Solution**:
1. Start from \( x = \frac{1}{2}y - 6 \).
2. Solve for \( y \):
\[
y = 2x + 12
\]
#### Problem 11: Linear Function Analysis
**Selected values of a linear function \( f(x) \) are provided in the table below.**
\[
\begin{array}{c|cccccc}
x & -8 & -2 & 4 & 12 & 14 & 18 \\
\hline
f(x) & -33 & -12 & 9 & k & 44 & 58 \\
\end{array}
\]
- Find the value of \( k \). Explain how you found your answer.
**Solution**:
1. Determine the slope \( m \) using any two complete pairs, such as \( (-8, -33) \) and \( (4, 9) \):
\[
m = \frac{9 - (-33)}{4 - (-8)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff874369a-1501-4297-ae95-de7ae7d9f557%2Fecc741af-5204-4d80-81d0-f2ab3de2b9d9%2Fnjguali_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Analyzing Linear Functions and Their Inverses
#### Problem 10: Linear Graph Analysis
**Given Instructions**:
- Analyze the linear graph provided and answer the following questions.
**(a) Equation of the Line**:
- Write the equation of the line in \( y = mx + b \) form.
**Solution**:
1. By observing the graph, determine the slope (\(m\)) and y-intercept (\(b\)).
2. From the calculations shown, the slope \(m\) is given by:
\[
m = \frac{6 - 0}{0 + 6} = \frac{1}{2}
\]
3. The equation, considering the y-intercept (-6), is:
\[
y = \frac{1}{2}x - 6
\]
**(b) Inverse Graph**:
- Create a graph of this linear function's inverse on the same set of graph paper.
**Explanation**:
- To graph the inverse, \( y = \frac{1}{2}x - 6 \) should be swapped to \( x = \frac{1}{2}y - 6 \) and solved for \( y \).
**(c) Equation of the Inverse**:
- Determine the equation of the inverse.
**Solution**:
1. Start from \( x = \frac{1}{2}y - 6 \).
2. Solve for \( y \):
\[
y = 2x + 12
\]
#### Problem 11: Linear Function Analysis
**Selected values of a linear function \( f(x) \) are provided in the table below.**
\[
\begin{array}{c|cccccc}
x & -8 & -2 & 4 & 12 & 14 & 18 \\
\hline
f(x) & -33 & -12 & 9 & k & 44 & 58 \\
\end{array}
\]
- Find the value of \( k \). Explain how you found your answer.
**Solution**:
1. Determine the slope \( m \) using any two complete pairs, such as \( (-8, -33) \) and \( (4, 9) \):
\[
m = \frac{9 - (-33)}{4 - (-8)}
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