Select The Correct Answer.What Is The End Behavior Of This Radical Function? F ( X ) = − 2 X + 7 3 F(x) = -2 \sqrt[3]{x+7} F ( X ) = − 2 3 X + 7 ​ A. As X X X Approaches Negative Infinity, F ( X F(x F ( X ] Approaches 0. B. As X X X Approaches Positive Infinity,

Introduction

Radical functions, also known as root functions, are a type of mathematical function that involves the extraction of a root of a number. In this article, we will explore the end behavior of a specific radical function, $f(x) = -2 \sqrt[3]{x+7}$, and determine the correct answer among the given options.

What is End Behavior?

End behavior refers to the behavior of a function as the input values approach positive or negative infinity. In other words, it describes what happens to the function as the input values get arbitrarily large or small. Understanding the end behavior of a function is crucial in mathematics, as it helps us predict the behavior of the function in different regions of its domain.

The Given Function

The given function is $f(x) = -2 \sqrt[3]{x+7}$. This function involves a cube root, which is a type of radical. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this case, the cube root of $x+7$ is raised to the power of $-2$, which means that the function will approach negative infinity as $x$ approaches positive infinity.

Option A: As $x$ approaches negative infinity, $f(x)$ approaches 0

To determine if this option is correct, we need to evaluate the function as $x$ approaches negative infinity. As $x$ gets arbitrarily large in the negative direction, the value of $x+7$ will approach negative infinity. Since the cube root of a negative number is negative, the value of $\sqrt[3]{x+7}$ will approach negative infinity. However, the function $f(x) = -2 \sqrt[3]{x+7}$ involves a negative sign, which will flip the sign of the function. Therefore, as $x$ approaches negative infinity, $f(x)$ will approach positive infinity, not 0.

Option B: As $x$ approaches positive infinity, $f(x)$ approaches 0

To determine if this option is correct, we need to evaluate the function as $x$ approaches positive infinity. As $x$ gets arbitrarily large in the positive direction, the value of $x+7$ will approach positive infinity. Since the cube root of a positive number is positive, the value of $\sqrt[3]{x+7}$ will approach positive infinity. However, the function $f(x) = -2 \sqrt[3]{x+7}$ involves a negative sign, which will flip the sign of the function. Therefore, as $x$ approaches positive infinity, $f(x)$ will approach negative infinity, not 0.

Conclusion

In conclusion, the correct answer is not among the given options. As $x$ approaches negative infinity, $f(x)$ will approach positive infinity, not 0. As $x$ approaches positive infinity, $f(x)$ will approach negative infinity, not 0. Therefore, the end behavior of the function $f(x) = -2 \sqrt[3]{x+7}$ is not as described in the given options.

Understanding the End Behavior of Radical Functions: Key Takeaways

• End behavior refers to the behavior of a function as the input values approach positive or negative infinity.
• The end behavior of a function can be determined by evaluating the function as the input values approach positive or negative infinity.
• Radical functions, such as the cube root function, can exhibit different end behavior depending on the sign of the input values.
• The end behavior of a function can be used to predict the behavior of the function in different regions of its domain.

References

• [1] "Radical Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/radical.html
• [2] "End Behavior of Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/endbehav.htm

Additional Resources

• Khan Academy: Radical Functions
• Mathway: Radical Functions
• Wolfram Alpha: Radical Functions
  Q&A: Understanding the End Behavior of Radical Functions =====================================================

Introduction

In our previous article, we explored the end behavior of a specific radical function, $f(x) = -2 \sqrt[3]{x+7}$. We determined that the end behavior of this function is not as described in the given options. In this article, we will answer some frequently asked questions about the end behavior of radical functions.

Q: What is the end behavior of a radical function?

A: The end behavior of a radical function refers to the behavior of the function as the input values approach positive or negative infinity. In other words, it describes what happens to the function as the input values get arbitrarily large or small.

Q: How do I determine the end behavior of a radical function?

A: To determine the end behavior of a radical function, you need to evaluate the function as the input values approach positive or negative infinity. This can be done by substituting large positive or negative values for the input variable and observing the behavior of the function.

Q: What is the difference between the end behavior of a radical function and the end behavior of a polynomial function?

A: The end behavior of a radical function is different from the end behavior of a polynomial function. While polynomial functions tend to approach positive or negative infinity as the input values approach positive or negative infinity, radical functions can exhibit different end behavior depending on the sign of the input values.

Q: Can you give an example of a radical function with a specific end behavior?

A: Yes, consider the function $f(x) = \sqrt[3]{x+7}$. As $x$ approaches negative infinity, the value of $x+7$ will approach negative infinity, and the cube root of a negative number is negative. Therefore, as $x$ approaches negative infinity, $f(x)$ will approach negative infinity.

Q: How do I determine the end behavior of a function with multiple radical terms?

A: To determine the end behavior of a function with multiple radical terms, you need to evaluate the function as the input values approach positive or negative infinity. This can be done by substituting large positive or negative values for the input variable and observing the behavior of the function.

Q: Can you give an example of a function with multiple radical terms and a specific end behavior?

A: Yes, consider the function $f(x) = \sqrt[3]{x+7} + \sqrt[3]{x-3}$. As $x$ approaches negative infinity, the value of $x+7$ will approach negative infinity, and the cube root of a negative number is negative. Therefore, as $x$ approaches negative infinity, $f(x)$ will approach negative infinity.

Q: How do I use the end behavior of a function to predict its behavior in different regions of its domain?

A: The end behavior of a function can be used to predict its behavior in different regions of its domain. By evaluating the function as the input values approach positive or negative infinity, you can determine the behavior of the function in different regions of its domain.

Q: Can you give an example of how to use the end behavior of a function to predict its behavior in different regions of its domain?

A: Yes, consider the function $f(x) = \sqrt[3]{x+7}$. As $x$ approaches negative infinity, the value of $x+7$ will approach negative infinity, and the cube root of a negative number is negative. Therefore, as $x$ approaches negative infinity, $f(x)$ will approach negative infinity. This means that the function will be negative in the region where $x$ is negative.

Conclusion

In conclusion, the end behavior of a radical function refers to the behavior of the function as the input values approach positive or negative infinity. By evaluating the function as the input values approach positive or negative infinity, you can determine the end behavior of the function. The end behavior of a function can be used to predict its behavior in different regions of its domain.

Understanding the End Behavior of Radical Functions: Key Takeaways

• The end behavior of a radical function refers to the behavior of the function as the input values approach positive or negative infinity.
• The end behavior of a function can be determined by evaluating the function as the input values approach positive or negative infinity.
• Radical functions can exhibit different end behavior depending on the sign of the input values.
• The end behavior of a function can be used to predict its behavior in different regions of its domain.

References

• [1] "Radical Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/radical.html
• [2] "End Behavior of Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/endbehav.htm

Additional Resources

• Khan Academy: Radical Functions
• Mathway: Radical Functions
• Wolfram Alpha: Radical Functions