Express the function graphed on the axes below as a piecewise function. 10 8. 6.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![The image presents a graph with a piecewise function and asks to express this function in its mathematical form. Below is a detailed description and transcription based on the content of the image:
### Graph Description
The graph consists of two distinct linear segments:
1. **Left Segment:**
- **Line:** Starts at the open circle on the y-axis at \( y = 10 \), moving downward with a negative slope, until it reaches a point just before \( x = -2 \).
- **Line Equation:** This segment can be approximated as a line with a negative slope.
- **Domain:** \( x < -2 \).
2. **Right Segment:**
- **Line:** Begins at an open circle on the graph at \( x = 2, y = 3 \) and moves with a negative slope, going through \( x = 10 \).
- **Line Equation:** This segment appears as a straight line trending downward.
- **Domain:** \( x \geq 2 \).
### Piecewise Function Representation
The function \( f(x) \) can be expressed in a piecewise manner, capturing the behavior described above:
\[
f(x) =
\begin{cases}
-3x - 4 & \text{for } x < -2 \\
-x + 3 & \text{for } x \geq 2
\end{cases}
\]
### Explanation
- For the domain \( x < -2 \), the corresponding line segment has a formula possibly representing a negative slope that decreases as x decreases. The line starts from \( y = 10 \) on the y-axis.
- For the domain \( x \geq 2 \), the line follows a downward slope. It starts from a point above the axis and ends extending towards the negative y-values as x increases.
Note: The exact equations are approximations based on typical linear function format \((y = mx + b)\), matching the pattern observed on the graph where \(m\) is the slope and \(b\) is the y-intercept.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F607cfd73-341e-40b9-9da6-af1be2ce5002%2Fbd41a112-20b9-4814-a0df-48f4425c1add%2F26iu0q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image presents a graph with a piecewise function and asks to express this function in its mathematical form. Below is a detailed description and transcription based on the content of the image:
### Graph Description
The graph consists of two distinct linear segments:
1. **Left Segment:**
- **Line:** Starts at the open circle on the y-axis at \( y = 10 \), moving downward with a negative slope, until it reaches a point just before \( x = -2 \).
- **Line Equation:** This segment can be approximated as a line with a negative slope.
- **Domain:** \( x < -2 \).
2. **Right Segment:**
- **Line:** Begins at an open circle on the graph at \( x = 2, y = 3 \) and moves with a negative slope, going through \( x = 10 \).
- **Line Equation:** This segment appears as a straight line trending downward.
- **Domain:** \( x \geq 2 \).
### Piecewise Function Representation
The function \( f(x) \) can be expressed in a piecewise manner, capturing the behavior described above:
\[
f(x) =
\begin{cases}
-3x - 4 & \text{for } x < -2 \\
-x + 3 & \text{for } x \geq 2
\end{cases}
\]
### Explanation
- For the domain \( x < -2 \), the corresponding line segment has a formula possibly representing a negative slope that decreases as x decreases. The line starts from \( y = 10 \) on the y-axis.
- For the domain \( x \geq 2 \), the line follows a downward slope. It starts from a point above the axis and ends extending towards the negative y-values as x increases.
Note: The exact equations are approximations based on typical linear function format \((y = mx + b)\), matching the pattern observed on the graph where \(m\) is the slope and \(b\) is the y-intercept.
Expert Solution
Step 1
The stepwise function is the function that has different values for different values of x. The stepwise function may be discontinuous at some intervals.
The equation of a line can be obtained using two points form . Here, the points are .
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