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
Give a rule of the piecewise-defined function. Give the domain and the range.
![**Piecewise-Defined Functions**
---
**Problem Statement:**
Give a rule of the piecewise-defined function. Give the domain and the range.
**Graph Analysis:**
To fulfill this task, we analyze the provided graph on the right side of the screen. The graph shows a function f(x) represented by two line segments:
1. A leftward arrow starting from the point (2, 0) and extending horizontally to the left, which indicates constant function f(x) = 0 for \( x < 2 \).
2. A rightward arrow starting from the point (2, 2) and extending horizontally to the right, which indicates constant function f(x) = 2 for \( x \geq 2 \).
The horizontal arrows show that these lines continue indefinitely.
**Formulation of the Rule:**
Next, we will select the correct choice below and fill in the answer boxes within the choice given.
**Choices:**
A. \( f(x) = \begin{cases}
[\boxed{}] & \text{if } x < \boxed{} \\
[\boxed{}] & \text{if } x \geq \boxed{}
\end{cases}
\)
B. \( f(x) = \begin{cases}
[\boxed{}] & \text{if } x \leq \boxed{} \\
[\boxed{}] & \text{if } x > \boxed{}
\end{cases}
\)
In this case, we choose Option A and fill in the answer boxes:
- For \( x < 2 \), the function is constant at 0.
- For \( x \geq 2 \), the function is constant at 2.
So, the rule of the function is:
\[ f(x) = \begin{cases}
0 & \text{if } x < 2 \\
2 & \text{if } x \geq 2
\end{cases}
\]
**Domain and Range:**
- **Domain:** The domain of the function includes all real numbers since the function is defined for all \( x < 2 \) and \( x \geq 2 \), so the domain is \( (-\infty, \infty) \).
- **Range:** The range of the function includes the values that the function outputs, which are](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd645a57e-8fec-4f58-b186-7cff590ec41c%2F6184370e-266c-4dfc-9f15-8c1df7230466%2Fwfwbyle_processed.png&w=3840&q=75)
Transcribed Image Text:**Piecewise-Defined Functions**
---
**Problem Statement:**
Give a rule of the piecewise-defined function. Give the domain and the range.
**Graph Analysis:**
To fulfill this task, we analyze the provided graph on the right side of the screen. The graph shows a function f(x) represented by two line segments:
1. A leftward arrow starting from the point (2, 0) and extending horizontally to the left, which indicates constant function f(x) = 0 for \( x < 2 \).
2. A rightward arrow starting from the point (2, 2) and extending horizontally to the right, which indicates constant function f(x) = 2 for \( x \geq 2 \).
The horizontal arrows show that these lines continue indefinitely.
**Formulation of the Rule:**
Next, we will select the correct choice below and fill in the answer boxes within the choice given.
**Choices:**
A. \( f(x) = \begin{cases}
[\boxed{}] & \text{if } x < \boxed{} \\
[\boxed{}] & \text{if } x \geq \boxed{}
\end{cases}
\)
B. \( f(x) = \begin{cases}
[\boxed{}] & \text{if } x \leq \boxed{} \\
[\boxed{}] & \text{if } x > \boxed{}
\end{cases}
\)
In this case, we choose Option A and fill in the answer boxes:
- For \( x < 2 \), the function is constant at 0.
- For \( x \geq 2 \), the function is constant at 2.
So, the rule of the function is:
\[ f(x) = \begin{cases}
0 & \text{if } x < 2 \\
2 & \text{if } x \geq 2
\end{cases}
\]
**Domain and Range:**
- **Domain:** The domain of the function includes all real numbers since the function is defined for all \( x < 2 \) and \( x \geq 2 \), so the domain is \( (-\infty, \infty) \).
- **Range:** The range of the function includes the values that the function outputs, which are
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