Given The Function $f(x)=-(x+2)^3-4$, What Is The Domain And Range?

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Introduction

In mathematics, the domain and range of a function are crucial concepts that help us understand the behavior and properties of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of the given cubic function $f(x)=-(x+2)^3-4$.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given cubic function $f(x)=-(x+2)^3-4$, we can see that there are no restrictions on the input values (x-values). This means that the domain of the function is all real numbers, denoted as $(-\infty, \infty)$.

Domain: (-\infty, \infty)

Range of the Function

The range of a function is the set of all possible output values that the function can produce. To find the range of the given cubic function, we need to analyze its behavior. Since the function is a cubic function, it can have a maximum of three turning points (local maxima or minima). The given function is a cubic function with a negative leading coefficient, which means that it will have a local maximum and a local minimum.

To find the local maximum and minimum, we need to find the critical points of the function. The critical points are the points where the derivative of the function is equal to zero or undefined. Let's find the derivative of the function:

f′(x)=−3(x+2)2f'(x)=-3(x+2)^2

Now, let's find the critical points by setting the derivative equal to zero:

−3(x+2)2=0-3(x+2)^2=0

x+2=0x+2=0

x=−2x=-2

So, the critical point is x = -2. Now, let's evaluate the function at the critical point:

f(−2)=−((−2)+2)3−4f(-2)=-( (-2)+2)^3-4

f(−2)=−(−4)3−4f(-2)=-(-4)^3-4

f(−2)=64−4f(-2)=64-4

f(−2)=60f(-2)=60

Since the function is a cubic function, it will have a local maximum and a local minimum. The local maximum will occur at the critical point x = -2, and the local minimum will occur at the endpoints of the domain. Since the domain is all real numbers, the endpoints are x = -∞ and x = ∞.

Now, let's evaluate the function at the endpoints:

f(−∞)=−(−∞+2)3−4f(-\infty)=-( -\infty+2)^3-4

f(−∞)=∞f(-\infty)=\infty

f(∞)=−(∞+2)3−4f(\infty)=-( \infty+2)^3-4

f(∞)=−∞f(\infty)=-\infty

So, the local maximum is f(-2) = 60, and the local minimum is f(-∞) = ∞ and f(∞) = -∞.

Since the function is a cubic function, it will have a range that includes all real numbers. However, the local maximum and minimum will restrict the range. The range will be all real numbers less than or equal to 60 and greater than or equal to -∞.

Range: (-\infty, 60]

Conclusion

In conclusion, the domain of the given cubic function $f(x)=-(x+2)^3-4$ is all real numbers, denoted as $(-\infty, \infty)$. The range of the function is all real numbers less than or equal to 60 and greater than or equal to -∞, denoted as $(-\infty, 60]$. The local maximum and minimum of the function are f(-2) = 60 and f(-∞) = ∞ and f(∞) = -∞, respectively.

References

  • [1] Calculus, James Stewart, 8th edition
  • [2] Algebra and Trigonometry, Michael Sullivan, 4th edition
  • [3] Precalculus, Michael Sullivan, 4th edition

Future Work

In the future, we can explore other types of functions and their domains and ranges. We can also analyze the behavior of the functions and their critical points. Additionally, we can explore the applications of functions in real-world problems.

Code

import numpy as np

def f(x):
    return -(x+2)**3-4

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

import matplotlib.pyplot as plt

plt.plot(x, y)
plt.title('Cubic Function')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.axhline(60, color='r', linestyle='--')
plt.axhline(-np.inf, color='g', linestyle='--')
plt.axhline(np.inf, color='g', linestyle='--')
plt.show()
```<br/>
# Domain and Range of a Cubic Function: Q&A
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## Introduction
---------------

In our previous article, we explored the domain and range of the given cubic function $f(x)=-(x+2)^3-4$. We found that the domain of the function is all real numbers, denoted as $(-\infty, \infty)$, and the range of the function is all real numbers less than or equal to 60 and greater than or equal to -∞, denoted as $(-\infty, 60]$. In this article, we will answer some frequently asked questions about the domain and range of the function.

## Q&A
------

### Q: What is the domain of the function?

A: The domain of the function is all real numbers, denoted as $(-\infty, \infty)$.

### Q: What is the range of the function?

A: The range of the function is all real numbers less than or equal to 60 and greater than or equal to -∞, denoted as $(-\infty, 60]$.

### Q: Why is the domain all real numbers?

A: The domain is all real numbers because there are no restrictions on the input values (x-values) of the function.

### Q: Why is the range all real numbers less than or equal to 60 and greater than or equal to -∞?

A: The range is all real numbers less than or equal to 60 and greater than or equal to -∞ because the function has a local maximum at x = -2 and the local minimum at x = -∞ and x = ∞.

### Q: What is the local maximum of the function?

A: The local maximum of the function is f(-2) = 60.

### Q: What is the local minimum of the function?

A: The local minimum of the function is f(-∞) = ∞ and f(∞) = -∞.

### Q: How can I find the domain and range of a function?

A: To find the domain and range of a function, you need to analyze the behavior of the function and its critical points. You can use calculus to find the derivative of the function and set it equal to zero to find the critical points.

### Q: What are the applications of functions in real-world problems?

A: Functions have many applications in real-world problems, such as modeling population growth, predicting stock prices, and optimizing business decisions.

## Conclusion
----------

In conclusion, the domain and range of the given cubic function $f(x)=-(x+2)^3-4$ are all real numbers and all real numbers less than or equal to 60 and greater than or equal to -∞, respectively. We answered some frequently asked questions about the domain and range of the function and provided some tips on how to find the domain and range of a function.

## References
----------

* [1] Calculus, James Stewart, 8th edition
* [2] Algebra and Trigonometry, Michael Sullivan, 4th edition
* [3] Precalculus, Michael Sullivan, 4th edition

## Future Work
--------------

In the future, we can explore other types of functions and their domains and ranges. We can also analyze the behavior of the functions and their critical points. Additionally, we can explore the applications of functions in real-world problems.

## Code
------

```python
import numpy as np

def f(x):
    return -(x+2)**3-4

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

import matplotlib.pyplot as plt

plt.plot(x, y)
plt.title('Cubic Function')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.axhline(60, color='r', linestyle='--')
plt.axhline(-np.inf, color='g', linestyle='--')
plt.axhline(np.inf, color='g', linestyle='--')
plt.show()

Additional Resources

  • [1] Khan Academy: Functions
  • [2] MIT OpenCourseWare: Calculus
  • [3] Wolfram Alpha: Functions