Explain How To Graph The Given Piecewise-defined Function. Be Sure To Specify The Type Of Endpoint Each Piece Of The Function Will Have And Why.$f(x)=\left{\begin{array}{ll} -x+3, & X\ \textless \ 2 \ 3, & 2 \leq X\ \textless \ 4 \ 4-2x, & X

Understanding Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by multiple sub-functions, each of which is 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. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena that exhibit different behaviors in different regions.

The Given Piecewise-Defined Function

The given piecewise-defined function is:

$$f(x)=\left\{\begin{array}{ll} -x+3, & x\ \textless \ 2 \\ 3, & 2 \leq x\ \textless \ 4 \\ 4-2x, & x \end{array}\right.$$

This function has three different sub-functions, each of which is applied to a specific interval of the domain. The first sub-function, $-x+3$, is applied to the interval $x<2$. The second sub-function, $3$, is applied to the interval $2\leq x<4$. The third sub-function, $4-2x$, is applied to the interval $x\geq 4$.

Graphing the Piecewise-Defined Function

To graph the piecewise-defined function, we need to graph each of the sub-functions separately and then combine them to form the final graph.

Graphing the First Sub-Function

The first sub-function is $-x+3$, which is a linear function with a slope of $-1$ and a y-intercept of $3$. 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 matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 2, 400)

y = -x + 3

plt.plot(x, y, label='f(x) = -x + 3')
plt.title('Graph of the First Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Graphing the Second Sub-Function

The second sub-function is $3$, which is a constant function. To graph this function, we can simply plot a horizontal line at $y=3$.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(2, 4, 400)

y = 3

plt.plot(x, y * np.ones(len(x)), label='f(x) = 3')
plt.title('Graph of the Second Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Graphing the Third Sub-Function

The third sub-function is $4-2x$, which is a linear function with a slope of $-2$ and a y-intercept of $4$. 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 matplotlib.pyplot as plt
import numpy as np

x = np.linspace(4, 10, 400)

y = 4 - 2 * x

plt.plot(x, y, label='f(x) = 4 - 2x')
plt.title('Graph of the Third Sub-Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Combining the Sub-Functions

To combine the sub-functions, we need to plot each of them separately and then combine them to form the final graph.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-10, 10, 400)

y1 = -x + 3
y2 = 3 * np.ones(len(x))
y3 = 4 - 2 * x

plt.plot(x, y1, label='f(x) = -x + 3')
plt.plot(x, y2, label='f(x) = 3')
plt.plot(x, y3, label='f(x) = 4 - 2x')
plt.title('Graph of the Piecewise-Defined Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Conclusion

Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each of which is 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 each of the sub-functions separately and then combine them to form the final graph.

Q: What are the different types of endpoint each piece of the function will have?

A: The different types of endpoint each piece of the function will have are:

• Open endpoint: This is an endpoint that is not included in the domain of the function. For example, the function $f(x) = x^2$ has an open endpoint at $x = 0$.
• Closed endpoint: This is an endpoint that is included in the domain of the function. For example, the function $f(x) = x^2$ has a closed endpoint at $x = 1$.
• Half-open endpoint: This is an endpoint that is included in the domain of the function, but only for one of the sub-functions. For example, the function $f(x) = \begin{cases} x^2 & x < 1 \\ 2x & x \geq 1 \end{cases}$ has a half-open endpoint at $x = 1$.

Q: How do I determine the type of endpoint each piece of the function will have?

A: To determine the type of endpoint each piece of the function will have, you need to examine the domain of each sub-function and the values of the function at the endpoints.

Q: What are some common types of piecewise-defined functions?

A: Some common types of piecewise-defined functions include:

• Step functions: These are functions that have a constant value for a certain interval of the domain, and then jump to a different constant value at the next interval.
• Piecewise-linear functions: These are functions that have a linear function for a certain interval of the domain, and then jump to a different linear function at the next interval.
• Piecewise-polynomial functions: These are functions that have a polynomial function for a certain interval of the domain, and then jump to a different polynomial function at the next interval.

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

A: To graph a piecewise-defined function using Python, you can use the matplotlib library. Here is an example of how to graph a piecewise-defined function using Python:

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-10, 10, 400)

y1 = -x + 3
y2 = 3 * np.ones(len(x))
y3 = 4 - 2 * x

plt.plot(x, y1, label='f(x) = -x + 3')
plt.plot(x, y2, label='f(x) = 3')
plt.plot(x, y3, label='f(x) = 4 - 2x')
plt.title('Graph of the Piecewise-Defined Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.show()

Q: What are some common applications of piecewise-defined functions?

A: Some common applications of piecewise-defined functions include:

• Modeling real-world phenomena: Piecewise-defined functions can be used to model real-world phenomena that exhibit different behaviors in different regions.
• Solving optimization problems: Piecewise-defined functions can be used to solve optimization problems that involve different objective functions in different regions.
• Analyzing data: Piecewise-defined functions can be used to analyze data that exhibits different patterns in different regions.

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

A: To determine the domain of a piecewise-defined function, you need to examine the domain of each sub-function and the values of the function at the endpoints.

Q: What are some common mistakes to avoid when graphing piecewise-defined functions?

A: Some common mistakes to avoid when graphing piecewise-defined functions include:

• Not including the endpoints: Make sure to include the endpoints of each sub-function in the graph.
• Not labeling the sub-functions: Make sure to label each sub-function in the graph.
• Not using a consistent scale: Make sure to use a consistent scale for the x and y axes.