Graph The Piecewise-defined Function:$f(x)=\begin{cases} -1-x & \text{if } X \leq 2 \\ -5+2x & \text{if } X \ \textgreater \ 2 \end{cases}$Choose The Correct Graph Below:A. B. C. D.

Feb 27, 2025 by ADMIN 191 views

**Graphing Piecewise-Defined Functions: A Step-by-Step Guide**

Introduction

In mathematics, a piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will explore how to graph a piecewise-defined function, using the given function $f(x)=\begin{cases} -1-x & \text{if } x \leq 2 \\ -5+2x & \text{if } x \ \textgreater \ 2 \end{cases}$ as an example.

Understanding Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function is typically written in the form:

f(x)=\begin{cases} f_1(x) & \text{if } x \in D_1 \\ f_2(x) & \text{if } x \in D_2 \\ \vdots & \text{if } x \in D_n \end{cases}

where $f_1(x), f_2(x), \ldots, f_n(x)$ are the sub-functions, and $D_1, D_2, \ldots, D_n$ are the intervals of the domain.

Graphing the Piecewise-Defined Function

To graph the piecewise-defined function $f(x)=\begin{cases} -1-x & \text{if } x \leq 2 \\ -5+2x & \text{if } x \ \textgreater \ 2 \end{cases}$ , we need to graph the two sub-functions separately and then combine them.

Graphing the First Sub-Function

The first sub-function is $f_1(x) = -1 - x$ , which is a linear function with a slope of -1 and a y-intercept of -1. To graph this function, we can use the slope-intercept form of a linear function, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

import numpy as np
import matplotlib.pyplot as plt

# Define the x-values
x = np.linspace(-5, 2, 100)

# Define the y-values
y = -1 - x

# Plot the function
plt.plot(x, y)
plt.title('Graph of the First Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Graphing the Second Sub-Function

The second sub-function is $f_2(x) = -5 + 2x$ , which is also a linear function with a slope of 2 and a y-intercept of -5. To graph this function, we can use the same method as before.

import numpy as np
import matplotlib.pyplot as plt

# Define the x-values
x = np.linspace(2, 5, 100)

# Define the y-values
y = -5 + 2 * x

# Plot the function
plt.plot(x, y)
plt.title('Graph of the Second Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

Combining the Two Sub-Functions

To combine the two sub-functions, we need to graph the first sub-function for $x \leq 2$ and the second sub-function for $x > 2$ . We can use the numpy library to create an array of x-values that satisfy both conditions.

import numpy as np
import matplotlib.pyplot as plt

# Define the x-values
x = np.linspace(-5, 5, 100)

# Define the y-values for the first sub-function
y1 = np.where(x <= 2, -1 - x, np.nan)

# Define the y-values for the second sub-function
y2 = np.where(x > 2, -5 + 2 * x, np.nan)

# Plot the function
plt.plot(x, y1, label='First Sub-Function')
plt.plot(x, y2, label='Second Sub-Function')
plt.title('Graph of the Piecewise-Defined Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Conclusion

In this article, we have seen how to graph a piecewise-defined function using the given function $f(x)=\begin{cases} -1-x & \text{if } x \leq 2 \\ -5+2x & \text{if } x \ \textgreater \ 2 \end{cases}$ as an example. We have graphed the two sub-functions separately and then combined them to obtain the final graph. This method can be applied to any piecewise-defined function, and it provides a clear and concise way to visualize the behavior of the function over different intervals of the domain.

Discussion

The graph of a piecewise-defined function can be used to model real-world phenomena that exhibit different behaviors in different regions. For example, the graph of a piecewise-defined function can be used to model the behavior of a population that grows at a constant rate for a certain period of time and then suddenly changes to a different growth rate.

Choosing the Correct Graph

Based on the graph we have obtained, we can choose the correct graph from the options provided. The correct graph is the one that matches the graph we have obtained.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to graph a piecewise-defined function using the given function $f(x)=\begin{cases} -1-x & \text{if } x \leq 2 \\ -5+2x & \text{if } x \ \textgreater \ 2 \end{cases}$ as an example. In this article, we will answer some frequently asked questions about graphing piecewise-defined functions.

Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I graph a piecewise-defined function?

A: To graph a piecewise-defined function, you need to graph the sub-functions separately and then combine them. You can use the numpy library to create an array of x-values that satisfy both conditions.

Q: What is the difference between a piecewise-defined function and a continuous function?

A: A continuous function is a function that can be graphed without any breaks or gaps. A piecewise-defined function, on the other hand, is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: Can I use a piecewise-defined function to model real-world phenomena?

A: Yes, you can use a piecewise-defined function to model real-world phenomena that exhibit different behaviors in different regions.

Q: How do I choose the correct graph from the options provided?

A: To choose the correct graph, you need to match the graph you have obtained with the options provided.

Q: What is the final answer?

A: The final answer is the graph that matches the graph you have obtained.

Q: Can I use a piecewise-defined function to solve problems in mathematics and science?

A: Yes, you can use a piecewise-defined function to solve problems in mathematics and science. Piecewise-defined functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions.

Q: How do I graph a piecewise-defined function using a graphing calculator?

A: To graph a piecewise-defined function using a graphing calculator, you need to enter the function in the calculator and then use the graphing function to obtain the graph.

Q: Can I use a piecewise-defined function to model population growth?

A: Yes, you can use a piecewise-defined function to model population growth. For example, you can use a piecewise-defined function to model the growth of a population that grows at a constant rate for a certain period of time and then suddenly changes to a different growth rate.

Q: How do I determine the intervals of the domain for a piecewise-defined function?

A: To determine the intervals of the domain for a piecewise-defined function, you need to examine the function and identify the intervals where the function is defined.

Q: Can I use a piecewise-defined function to model economic systems?

A: Yes, you can use a piecewise-defined function to model economic systems. For example, you can use a piecewise-defined function to model the behavior of an economy that exhibits different behaviors in different regions.

Conclusion

In this article, we have answered some frequently asked questions about graphing piecewise-defined functions. We have also provided examples of how to use piecewise-defined functions to model real-world phenomena. By understanding how to graph piecewise-defined functions, you can use them to solve problems in mathematics and science.

Discussion

Final Answer

The final answer is: