Graphing A Piecewise Defined FunctionWhich Graph Represents The Piecewise-defined Function $\[ F(x) = \begin{cases} -1.5x + 3.5, & \text{if } X \ \textless \ 2 \\ 4 + X, & \text{if } X \geq 2 \end{cases} \\]

Feb 27, 2025 by ADMIN 212 views

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. In other words, a piecewise defined function is a function that has different formulas for different parts of its domain. This type of function is commonly used to model real-world phenomena that exhibit different behaviors in different regions.

The Given Piecewise Defined Function

The given piecewise defined function is:

f(x) = \begin{cases} -1.5x + 3.5, & \text{if } x \ \textless \ 2 \\ 4 + x, & \text{if } x \geq 2 \end{cases}

This function has two sub-functions: $-1.5x + 3.5$ for $x < 2$ and $4 + x$ for $x \geq 2$ . To graph this function, we need to graph each sub-function separately and then combine them.

Graphing the First Sub-Function

The first sub-function is $-1.5x + 3.5$ . This is a linear function with a slope of $-1.5$ and a y-intercept of $3.5$ . 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(-10, 10, 400)

# Define the y values
y = -1.5*x + 3.5

# Create the plot
plt.plot(x, y)
plt.title('Graph of the First Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Graphing the Second Sub-Function

The second sub-function is $4 + x$ . This is also a linear function with a slope of $1$ and a y-intercept of $4$ . To graph this function, we can use the slope-intercept form of a linear function.

import numpy as np
import matplotlib.pyplot as plt

# Define the x values
x = np.linspace(-10, 10, 400)

# Define the y values
y = 4 + x

# Create the plot
plt.plot(x, y)
plt.title('Graph of the Second Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Graphing the Piecewise Defined Function

To graph the piecewise defined function, we need to graph each sub-function separately and then combine them. We can use the matplotlib library to create the graph.

import numpy as np
import matplotlib.pyplot as plt

# Define the x values
x = np.linspace(-10, 10, 400)

# Define the y values for the first sub-function
y1 = -1.5*x + 3.5

# Define the y values for the second sub-function
y2 = 4 + x

# Create the plot
plt.plot(x, y1, label='x < 2')
plt.plot(x, y2, label='x >= 2')
plt.title('Graph of the Piecewise Defined Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.legend()
plt.show()

Conclusion

In this article, we graphed a piecewise defined function using the matplotlib library. We first graphed each sub-function separately and then combined them to create the graph of the piecewise defined function. This type of function is commonly used to model real-world phenomena that exhibit different behaviors in different regions.

References

[1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
[2] "Graphing Piecewise Functions." Purplemath, purplemath.com/modules/piecewise.htm.

Discussion

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. In other words, a piecewise defined function is a function that has different formulas for different parts of its domain.

Q: How do I graph a piecewise defined function?

A: To graph a piecewise defined function, you need to graph each sub-function separately and then combine them. You can use the matplotlib library to create the graph.

Q: What are the different types of piecewise defined functions?

A: There are two main types of piecewise defined functions: linear and non-linear. Linear piecewise defined functions have a constant slope, while non-linear piecewise defined functions have a variable slope.

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

A: To determine the intervals for a piecewise defined function, you need to look at the function's domain and identify the points where the function changes its behavior. These points are called the "break points" or "transition points".

Q: What is the purpose of a piecewise defined function?

A: The purpose of a piecewise defined function is to model real-world phenomena that exhibit different behaviors in different regions. For example, a piecewise defined function can be used to model the cost of a product that changes depending on the quantity ordered.

Q: Can I use a piecewise defined function to model a real-world phenomenon that has multiple peaks or valleys?

A: Yes, you can use a piecewise defined function to model a real-world phenomenon that has multiple peaks or valleys. However, you will need to use multiple sub-functions to capture the different behaviors of the phenomenon.

Q: How do I graph a piecewise defined function with multiple sub-functions?

A: To graph a piecewise defined function with multiple sub-functions, you need to graph each sub-function separately and then combine them. You can use the matplotlib library to create the graph.

Q: Can I use a piecewise defined function to model a real-world phenomenon that has a discontinuity?

A: Yes, you can use a piecewise defined function to model a real-world phenomenon that has a discontinuity. However, you will need to use a sub-function that is defined at the point of discontinuity.

Q: How do I determine the sub-functions for a piecewise defined function?

A: To determine the sub-functions for a piecewise defined function, you need to look at the function's domain and identify the points where the function changes its behavior. These points are called the "break points" or "transition points".

Q: Can I use a piecewise defined function to model a real-world phenomenon that has a periodic behavior?

A: Yes, you can use a piecewise defined function to model a real-world phenomenon that has a periodic behavior. However, you will need to use a sub-function that is defined at the point of periodicity.

Q: How do I graph a piecewise defined function with a periodic behavior?

A: To graph a piecewise defined function with a periodic behavior, you need to graph each sub-function separately and then combine them. You can use the matplotlib library to create the graph.

Q: Can I use a piecewise defined function to model a real-world phenomenon that has a non-linear behavior?

A: Yes, you can use a piecewise defined function to model a real-world phenomenon that has a non-linear behavior. However, you will need to use a sub-function that is defined at the point of non-linearity.

Q: How do I graph a piecewise defined function with a non-linear behavior?

A: To graph a piecewise defined function with a non-linear behavior, you need to graph each sub-function separately and then combine them. You can use the matplotlib library to create the graph.

Conclusion

In this article, we answered some common questions about graphing piecewise defined functions. We covered topics such as the definition of a piecewise defined function, how to graph a piecewise defined function, and how to determine the sub-functions for a piecewise defined function. We also discussed how to graph a piecewise defined function with multiple sub-functions, a periodic behavior, and a non-linear behavior.

References

[1] "Piecewise Functions." Math Open Reference, mathopenref.com/piecewise.html.
[2] "Graphing Piecewise Functions." Purplemath, purplemath.com/modules/piecewise.htm.

Discussion

What is a piecewise defined function? How do you graph a piecewise defined function? Can you think of a real-world example of a piecewise defined function? How would you determine the sub-functions for a piecewise defined function?