Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Write the equation shown in the graph.
![### Piecewise-Defined Function Graph Explanation
The image contains a graph and a piecewise function, highlighting different expressions based on the domain of the variable \( x \).
#### The Piecewise Function:
\[
f(x) =
\begin{cases}
|x + 3| & \text{for } -5 < x < -2 \\
\frac{1}{2} |x| & \text{for } -2 < x < 2 \\
(x - 2)^2 & \text{for } x \ge 2
\end{cases}
\]
### Graph Analysis:
1. **Domain \(-5 < x < -2\):**
- **Equation:** \( |x + 3| \)
- **Behavior:** The graph in this section represents the absolute value function of \( x + 3 \), which shows a 'V' shape. For the interval \(-5 < x < -2\), the function rises as it approaches \( x = -2 \).
2. **Domain \(-2 < x < 2\):**
- **Equation:** \( \frac{1}{2} |x| \)
- **Behavior:** In this range, the graph displays a linear function scaled down by a factor of \(\frac{1}{2}\). The line is symmetrical about the y-axis and is a series of two linear segments with a gentler slope.
3. **Domain \(x \ge 2\):**
- **Equation:** \( (x - 2)^2 \)
- **Behavior:** For this domain, the function represents a parabola shifted to the right by 2 units. It opens upwards, starting from point \( (2, 0) \) on the graph, demonstrating the squaring of \((x - 2)\).
### Graphical Details:
- **Axes:** Labeled as \( x \) and \( y \) with increment markings of 5 units.
- **Points and Lines:**
- There is an open circle at \( (2, 0) \) indicating that the point is not included for the function \((x - 2)^2\) when transitioning from the previous segment.
- Solid lines indicate included values in the domains described above.
This representation highlights how different algebraic expressions form a single piecewise function over distinct intervals of \( x \). The](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9984d7ae-1a86-419a-b4b6-309f0e42346f%2F420f46e9-29e5-4d7c-ae03-07df4cfe8481%2Fi8laris_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise-Defined Function Graph Explanation
The image contains a graph and a piecewise function, highlighting different expressions based on the domain of the variable \( x \).
#### The Piecewise Function:
\[
f(x) =
\begin{cases}
|x + 3| & \text{for } -5 < x < -2 \\
\frac{1}{2} |x| & \text{for } -2 < x < 2 \\
(x - 2)^2 & \text{for } x \ge 2
\end{cases}
\]
### Graph Analysis:
1. **Domain \(-5 < x < -2\):**
- **Equation:** \( |x + 3| \)
- **Behavior:** The graph in this section represents the absolute value function of \( x + 3 \), which shows a 'V' shape. For the interval \(-5 < x < -2\), the function rises as it approaches \( x = -2 \).
2. **Domain \(-2 < x < 2\):**
- **Equation:** \( \frac{1}{2} |x| \)
- **Behavior:** In this range, the graph displays a linear function scaled down by a factor of \(\frac{1}{2}\). The line is symmetrical about the y-axis and is a series of two linear segments with a gentler slope.
3. **Domain \(x \ge 2\):**
- **Equation:** \( (x - 2)^2 \)
- **Behavior:** For this domain, the function represents a parabola shifted to the right by 2 units. It opens upwards, starting from point \( (2, 0) \) on the graph, demonstrating the squaring of \((x - 2)\).
### Graphical Details:
- **Axes:** Labeled as \( x \) and \( y \) with increment markings of 5 units.
- **Points and Lines:**
- There is an open circle at \( (2, 0) \) indicating that the point is not included for the function \((x - 2)^2\) when transitioning from the previous segment.
- Solid lines indicate included values in the domains described above.
This representation highlights how different algebraic expressions form a single piecewise function over distinct intervals of \( x \). The
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