Give a rule of the​ piecewise-defined function. Give the domain and the range.

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,...
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
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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