Determine if the function is continuous or discontinuous, describe the end behavior, and then determine the intervals over which each function is increasing, decreasing, and constant. Write "none" if there is no interval. mi' [x+3, x50 f(x) 3, (2x+1, x>2 (0,3) √-3-2 75 01 0.< x≤2 £12. (2,5) (2,3) he Continuous/Discontinuous End Behavior: x→ -00, y➜ x→ co, y→ Increasing: Decreasing: Constant:
Determine if the function is continuous or discontinuous, describe the end behavior, and then determine the intervals over which each function is increasing, decreasing, and constant. Write "none" if there is no interval. mi' [x+3, x50 f(x) 3, (2x+1, x>2 (0,3) √-3-2 75 01 0.< x≤2 £12. (2,5) (2,3) he Continuous/Discontinuous End Behavior: x→ -00, y➜ x→ co, y→ Increasing: Decreasing: Constant:
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
![**Function Analysis**
This resource explores the properties of a piecewise function, detailing its continuity, end behavior, and intervals of increase, decrease, and constancy.
**Piecewise Function Definition:**
- \( f(x) = \begin{cases}
x + 3, & x \leq 0 \\
3, & 0 < x \leq 2 \\
2x + 1, & x > 2
\end{cases} \)
**Graph Description:**
- The graph includes three segments:
- A line from (-3, 0) to (0, 3) showing \( y = x + 3 \) for \( x \leq 0 \).
- A constant line \( y = 3 \) for \( 0 < x \leq 2 \), including a filled circle at (0, 3) and an open circle at (2, 3).
- A line \( y = 2x + 1 \) for \( x > 2 \) starting from the open circle at (2, 5).
**Analysis:**
- **Continuous/Discontinuous:**
- The function is discontinuous at \( x = 0 \) and \( x = 2 \).
- **End Behavior:**
- As \( x \to -\infty \), \( y \to -\infty \).
- As \( x \to \infty \), \( y \to \infty \).
- **Intervals:**
- **Increasing:** \( (-\infty, 0) \) and \( (2, \infty) \)
- **Decreasing:** None
- **Constant:** \( (0, 2] \)
Use this guide to understand how the function behaves across its domain, observing its continuity and changes in value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bafa714-33cc-47c4-a9a5-96cfac78ec18%2F09ba1327-b2ec-49db-a79f-46b2da95ae79%2Fdj334yo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Function Analysis**
This resource explores the properties of a piecewise function, detailing its continuity, end behavior, and intervals of increase, decrease, and constancy.
**Piecewise Function Definition:**
- \( f(x) = \begin{cases}
x + 3, & x \leq 0 \\
3, & 0 < x \leq 2 \\
2x + 1, & x > 2
\end{cases} \)
**Graph Description:**
- The graph includes three segments:
- A line from (-3, 0) to (0, 3) showing \( y = x + 3 \) for \( x \leq 0 \).
- A constant line \( y = 3 \) for \( 0 < x \leq 2 \), including a filled circle at (0, 3) and an open circle at (2, 3).
- A line \( y = 2x + 1 \) for \( x > 2 \) starting from the open circle at (2, 5).
**Analysis:**
- **Continuous/Discontinuous:**
- The function is discontinuous at \( x = 0 \) and \( x = 2 \).
- **End Behavior:**
- As \( x \to -\infty \), \( y \to -\infty \).
- As \( x \to \infty \), \( y \to \infty \).
- **Intervals:**
- **Increasing:** \( (-\infty, 0) \) and \( (2, \infty) \)
- **Decreasing:** None
- **Constant:** \( (0, 2] \)
Use this guide to understand how the function behaves across its domain, observing its continuity and changes in value.
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