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Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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---
**Piecewise Function Definition:**
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
x + 1 & \text{for } x < -3 \\
-3 & \text{for } -3 \leq x \leq 2 \\
-4 & \text{for } x > 2
\end{cases}
\]
**Explanation:**
This is a piecewise function, which means that the function \( f(x) \) is defined by different expressions depending on the value of \( x \).
1. **For values of \( x \) less than -3:**
- The function \( f(x) \) is given by the expression \( x + 1 \).
2. **For values of \( x \) between -3 and 2 inclusive:**
- The function \( f(x) \) is constant and equals -3.
3. **For values of \( x \) greater than 2:**
- The function \( f(x) \) is constant and equals -4.
**Graphical Representation:**
A graph of this piecewise function would have three distinct parts:
1. A line with a slope of 1 (since the expression \( x + 1 \) is linear) for \( x < -3 \).
2. A horizontal line at \( y = -3 \) from \( x = -3 \) to \( x = 2 \).
3. Another horizontal line at \( y = -4 \) for \( x > 2 \).
**Important Points to Note:**
- At \( x = -3 \) and \( x = 2 \), there will be filled dots indicating that the value at those points is included, ensuring the function is well defined and continuous at these specific points of the interval.
---
This detailed breakdown illustrates how to interpret and represent a piecewise function mathematically and graphically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0e15bb1-0e52-43c9-be1d-fbffb903a293%2F939e3249-3048-47a0-b27f-16f003656707%2Ff152c6v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription for the image, intended for use on an educational website:
---
**Piecewise Function Definition:**
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
x + 1 & \text{for } x < -3 \\
-3 & \text{for } -3 \leq x \leq 2 \\
-4 & \text{for } x > 2
\end{cases}
\]
**Explanation:**
This is a piecewise function, which means that the function \( f(x) \) is defined by different expressions depending on the value of \( x \).
1. **For values of \( x \) less than -3:**
- The function \( f(x) \) is given by the expression \( x + 1 \).
2. **For values of \( x \) between -3 and 2 inclusive:**
- The function \( f(x) \) is constant and equals -3.
3. **For values of \( x \) greater than 2:**
- The function \( f(x) \) is constant and equals -4.
**Graphical Representation:**
A graph of this piecewise function would have three distinct parts:
1. A line with a slope of 1 (since the expression \( x + 1 \) is linear) for \( x < -3 \).
2. A horizontal line at \( y = -3 \) from \( x = -3 \) to \( x = 2 \).
3. Another horizontal line at \( y = -4 \) for \( x > 2 \).
**Important Points to Note:**
- At \( x = -3 \) and \( x = 2 \), there will be filled dots indicating that the value at those points is included, ensuring the function is well defined and continuous at these specific points of the interval.
---
This detailed breakdown illustrates how to interpret and represent a piecewise function mathematically and graphically.
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