Suppose that the function f is defined, for all real numbers, as follows. -1 if x#0 f(x)= - 2. if x 0 Graph the function f. 3- 2+ 2. -1+ -2+ -3+
Suppose that the function f is defined, for all real numbers, as follows. -1 if x#0 f(x)= - 2. if x 0 Graph the function f. 3- 2+ 2. -1+ -2+ -3+
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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![Title: Graphing a Piecewise-Defined Function: Problem Type 1
---
**Introduction:**
In this problem, we are asked to graph a piecewise-defined function. These functions have different expressions based on the input value.
---
**Function Definition:**
The function \( f \) is defined for all real numbers \( x \), as follows:
\[
f(x) =
\begin{cases}
-1 & \text{if } x \neq 0 \\
2 & \text{if } x = 0
\end{cases}
\]
---
**Graph Explanation:**
- **Axes:** The graph is drawn on a standard Cartesian coordinate system with horizontal and vertical axes.
- **Graph of \( f \):**
- The line \( y = -1 \) is plotted for all \( x \neq 0 \). This is a horizontal line that extends across the graph except at \( x = 0 \).
- At \( x = 0 \), there is a distinct point plotted at \( y = 2 \).
- **Key Features:**
- There is an open circle on the line \( y = -1 \) where \( x = 0 \), indicating that this value is not included at \( x = 0 \).
- A filled circle represents the function value \( y = 2 \) precisely at \( x = 0 \), marking a point on the graph at (0, 2).
This illustrates a typical piecewise-defined function where a different rule applies at a specific \( x \)-value, which results in a jump or discontinuity on the graph.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90b823f6-a59b-43f1-924e-a48b4e93719a%2F4c981a27-2546-4326-aa14-9daa8ab0373e%2Fyjhty9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Graphing a Piecewise-Defined Function: Problem Type 1
---
**Introduction:**
In this problem, we are asked to graph a piecewise-defined function. These functions have different expressions based on the input value.
---
**Function Definition:**
The function \( f \) is defined for all real numbers \( x \), as follows:
\[
f(x) =
\begin{cases}
-1 & \text{if } x \neq 0 \\
2 & \text{if } x = 0
\end{cases}
\]
---
**Graph Explanation:**
- **Axes:** The graph is drawn on a standard Cartesian coordinate system with horizontal and vertical axes.
- **Graph of \( f \):**
- The line \( y = -1 \) is plotted for all \( x \neq 0 \). This is a horizontal line that extends across the graph except at \( x = 0 \).
- At \( x = 0 \), there is a distinct point plotted at \( y = 2 \).
- **Key Features:**
- There is an open circle on the line \( y = -1 \) where \( x = 0 \), indicating that this value is not included at \( x = 0 \).
- A filled circle represents the function value \( y = 2 \) precisely at \( x = 0 \), marking a point on the graph at (0, 2).
This illustrates a typical piecewise-defined function where a different rule applies at a specific \( x \)-value, which results in a jump or discontinuity on the graph.
---
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