Graph The Following Function On The Axes Provided:${ F(x) = \begin{cases} x+6 & \text{for} \quad -4 \ \textless \ X \ \textless \ -1 \ -2x+7 & \text{for} \quad -1 \ \textless \ X \leq 5 \end{cases} }$

Mar 11, 2025 by ADMIN 208 views

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

Introduction

Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will focus on graphing a specific piecewise function, $f(x) = \begin{cases} x+6 & \text{for} \quad -4 \ \textless \ x \ \textless \ -1 \\ -2x+7 & \text{for} \quad -1 \ \textless \ x \leq 5 \end{cases}$ , on the given axes.

Understanding Piecewise Functions

Before we dive into graphing the function, let's take a closer look at what piecewise functions are and how they work. A piecewise function is a function that is defined by multiple sub-functions, each of which is defined on a specific interval. In other words, a piecewise function is a function that has different formulas for different parts of its domain.

Graphing Piecewise Functions

Graphing a piecewise function involves graphing each sub-function separately and then combining the resulting graphs. To graph the function $f(x) = \begin{cases} x+6 & \text{for} \quad -4 \ \textless \ x \ \textless \ -1 \\ -2x+7 & \text{for} \quad -1 \ \textless \ x \leq 5 \end{cases}$ , we need to graph the two sub-functions separately and then combine the resulting graphs.

Graphing the First Sub-Function

The first sub-function is $f(x) = x+6$ for $-4 \ \textless \ x \ \textless \ -1$ . To graph this function, we can start by finding the x-intercept, which is the point where the function crosses the x-axis. To find the x-intercept, we set $f(x) = 0$ and solve for $x$ .

import numpy as np
def f(x):
return x + 6

x_intercept = -6
print(f"The x-intercept is {x_intercept}")

The x-intercept is $-6$ . Now that we have the x-intercept, we can graph the function by plotting the points $(x, f(x))$ for $x$ in the interval $-4 \ \textless \ x \ \textless \ -1$ .

Graphing the Second Sub-Function

The second sub-function is $f(x) = -2x+7$ for $-1 \ \textless \ x \leq 5$ . To graph this function, we can start by finding the x-intercept, which is the point where the function crosses the x-axis. To find the x-intercept, we set $f(x) = 0$ and solve for $x$ .

import numpy as np

def f(x):
return -2*x + 7

x_intercept = 3.5
print(f"The x-intercept is {x_intercept}")

The x-intercept is $3.5$ . Now that we have the x-intercept, we can graph the function by plotting the points $(x, f(x))$ for $x$ in the interval $-1 \ \textless \ x \leq 5$ .

Combining the Graphs

Now that we have graphed the two sub-functions separately, we can combine the resulting graphs to get the final graph of the piecewise function.

Graphing the Piecewise Function

The final graph of the piecewise function is a combination of the two sub-functions. The graph consists of two line segments, one for each sub-function. The first line segment is a straight line that passes through the points $(-4, 2)$ and $(-1, 5)$ . The second line segment is a straight line that passes through the points $(-1, 5)$ and $(5, -3)$ .

Conclusion

In this article, we graphed a piecewise function using Python. We started by defining the function and finding the x-intercepts of the two sub-functions. We then graphed the two sub-functions separately and combined the resulting graphs to get the final graph of the piecewise function. This article demonstrates the importance of piecewise functions in mathematics and provides a step-by-step guide on how to graph them.

References

[1] "Piecewise Functions" by Math Open Reference
[2] "Graphing Piecewise Functions" by Khan Academy

Code

import numpy as np
import matplotlib.pyplot as plt

def f1(x):
return x + 6
def f2(x):
return -2*x + 7

x = np.linspace(-4, 5, 100)

y1 = f1(x)
y2 = f2(x)

plt.figure()

plt.plot(x, y1, label='f(x) = x+6')

plt.plot(x, y2, label='f(x) = -2x+7')

plt.legend()

plt.show()

Introduction

In our previous article, we graphed a piecewise function using Python. We started by defining the function and finding the x-intercepts of the two sub-functions. We then graphed the two sub-functions separately and combined the resulting graphs to get the final graph of the piecewise function. In this article, we will answer some common questions about graphing piecewise functions.

Q: What is a piecewise function?

A piecewise function is a function that is defined by multiple sub-functions, each of which is defined on a specific interval. In other words, a piecewise function is a function that has different formulas for different parts of its domain.

Q: How do I graph a piecewise function?

To graph a piecewise function, you need to graph each sub-function separately and then combine the resulting graphs. You can use Python to graph the function by defining the function and finding the x-intercepts of the two sub-functions.

Q: What are the different types of piecewise functions?

There are two main types of piecewise functions: continuous and discontinuous. A continuous piecewise function is a function that is continuous on its entire domain, while a discontinuous piecewise function is a function that has a discontinuity at one or more points.

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

To determine the domain of a piecewise function, you need to identify the intervals on which each sub-function is defined. The domain of the function is the set of all x values for which the function is defined.

Q: Can I graph a piecewise function using a graphing calculator?

Yes, you can graph a piecewise function using a graphing calculator. Most graphing calculators have a built-in function for graphing piecewise functions.

Q: How do I find the x-intercepts of a piecewise function?

To find the x-intercepts of a piecewise function, you need to set the function equal to zero and solve for x. The x-intercepts are the points where the function crosses the x-axis.

Q: Can I graph a piecewise function using Python?

Yes, you can graph a piecewise function using Python. You can use the matplotlib library to graph the function.

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

To graph a piecewise function with multiple sub-functions, you need to graph each sub-function separately and then combine the resulting graphs. You can use Python to graph the function by defining the function and finding the x-intercepts of the two sub-functions.

Q: Can I graph a piecewise function with a discontinuity?

Yes, you can graph a piecewise function with a discontinuity. A discontinuity is a point where the function is not defined.

Q: How do I graph a piecewise function with a vertical asymptote?

A vertical asymptote is a line that the function approaches as x approaches a certain value. To graph a piecewise function with a vertical asymptote, you need to identify the point where the function approaches the asymptote and graph the function accordingly.

Q: Can I graph a piecewise function using a graphing software?

Yes, you can graph a piecewise function using a graphing software such as Desmos or GeoGebra.

Conclusion

In this article, we answered some common questions about graphing piecewise functions. We covered topics such as the definition of a piecewise function, how to graph a piecewise function, and how to determine the domain of a piecewise function. We also discussed how to graph a piecewise function using Python and a graphing calculator.

References

[1] "Piecewise Functions" by Math Open Reference
[2] "Graphing Piecewise Functions" by Khan Academy
[3] "Graphing Piecewise Functions using Python" by Real Python

Code

import numpy as np
import matplotlib.pyplot as plt

def f1(x):
return x + 6
def f2(x):
return -2*x + 7

x = np.linspace(-4, 5, 100)

y1 = f1(x)
y2 = f2(x)

plt.figure()

plt.plot(x, y1, label='f(x) = x+6')

plt.plot(x, y2, label='f(x) = -2x+7')

plt.legend()

plt.show()

This code will generate a plot of the piecewise function. The plot consists of two line segments, one for each sub-function. The first line segment is a straight line that passes through the points $(-4, 2)$ and $(-1, 5)$ . The second line segment is a straight line that passes through the points $(-1, 5)$ and $(5, -3)$ .