Select The Correct Answer.What Is The End Behavior Of The 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 Negative Infinity.B. As X X X Approaches Negative Infinity,

Introduction

Radical functions, also known as root functions, are a type of function that involves a variable or expression under a radical sign. In this article, we will focus on the end behavior of the radical function $f(x)=-2 \sqrt[3]{x+7}$. The end behavior of a function refers to the behavior of the function as the input or independent variable approaches positive or negative infinity.

What is the End Behavior of a Function?

The end behavior of a function is an important concept in mathematics, particularly in calculus and algebra. It helps us understand how a function behaves as the input or independent variable approaches positive or negative infinity. In other words, it tells us what happens to the function as the input gets very large or very small.

Radical Functions and Their End Behavior

Radical functions are a type of function that involves a variable or expression under a radical sign. The radical sign can be a square root, cube root, or any other root. The end behavior of a radical function depends on the degree of the root and the sign of the coefficient of the variable under the radical sign.

The Function $f(x)=-2 \sqrt[3]{x+7}$

The given function is $f(x)=-2 \sqrt[3]{x+7}$. This is a radical function with a cube root and a negative coefficient. To determine the end behavior of this function, we need to analyze the behavior of the cube root as the input or independent variable approaches positive or negative infinity.

As $x$ Approaches Negative Infinity

As $x$ approaches negative infinity, the expression under the cube root, $x+7$, also approaches negative infinity. Since the cube root of a negative number is negative, the cube root of $x+7$ will approach negative infinity. However, the negative coefficient of $-2$ will make the function approach positive infinity.

As $x$ Approaches Positive Infinity

As $x$ approaches positive infinity, the expression under the cube root, $x+7$, also approaches positive infinity. Since the cube root of a positive number is positive, the cube root of $x+7$ will approach positive infinity. However, the negative coefficient of $-2$ will make the function approach negative infinity.

Conclusion

In conclusion, the end behavior of the radical function $f(x)=-2 \sqrt[3]{x+7}$ is as follows:

• As $x$ approaches negative infinity, $f(x)$ approaches positive infinity.
• As $x$ approaches positive infinity, $f(x)$ approaches negative infinity.

Therefore, the correct answer is:

A. As $x$ approaches negative infinity, $f(x)$ approaches positive infinity.

Discussion

The end behavior of a function is an important concept in mathematics, particularly in calculus and algebra. It helps us understand how a function behaves as the input or independent variable approaches positive or negative infinity. In this article, we focused on the end behavior of the radical function $f(x)=-2 \sqrt[3]{x+7}$. We analyzed the behavior of the cube root as the input or independent variable approaches positive or negative infinity and determined the end behavior of the function.

References

• [1] "Calculus" by Michael Spivak
• [2] "Algebra" by Michael Artin
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

If you want to learn more about the end behavior of functions, I recommend checking out the following resources:

• Khan Academy: End Behavior of Functions
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: End Behavior of Functions
  Q&A: Understanding the End Behavior of Radical Functions =====================================================

Introduction

In our previous article, we discussed the end behavior of the radical function $f(x)=-2 \sqrt[3]{x+7}$. We analyzed the behavior of the cube root as the input or independent variable approaches positive or negative infinity and determined the end behavior of the function. 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 or independent variable approaches positive or negative infinity. It helps us understand how a function behaves as the input gets very large or very 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 analyze the behavior of the root as the input or independent variable approaches positive or negative infinity. You also need to consider the sign of the coefficient of the variable under the radical sign.

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

A: The end behavior of a radical function is different from the end behavior of a polynomial function. A polynomial function has a finite number of terms, while a radical function has an infinite number of terms. As a result, the end behavior of a radical function can be more complex than the end behavior of a polynomial function.

Q: Can the end behavior of a radical function be determined using the leading term?

A: Yes, the end behavior of a radical function can be determined using the leading term. The leading term is the term with the highest degree of the variable. If the leading term is positive, the function will approach positive infinity as the input or independent variable approaches positive infinity. If the leading term is negative, the function will approach negative infinity as the input or independent variable approaches positive infinity.

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

A: To determine the end behavior of a radical function with a negative coefficient, you need to analyze the behavior of the root as the input or independent variable approaches positive or negative infinity. You also need to consider the sign of the coefficient of the variable under the radical sign. If the coefficient is negative, the function will approach the opposite of the root as the input or independent variable approaches positive or negative infinity.

Q: Can the end behavior of a radical function be determined using a graph?

A: Yes, the end behavior of a radical function can be determined using a graph. By graphing the function, you can see how the function behaves as the input or independent variable approaches positive or negative infinity. This can help you determine the end behavior of the function.

Q: What are some common mistakes to avoid when determining the end behavior of a radical function?

A: Some common mistakes to avoid when determining the end behavior of a radical function include:

• Not considering the sign of the coefficient of the variable under the radical sign
• Not analyzing the behavior of the root as the input or independent variable approaches positive or negative infinity
• Not using the leading term to determine the end behavior of the function
• Not graphing the function to visualize the end behavior

Conclusion

In conclusion, determining the end behavior of a radical function requires careful analysis of the behavior of the root as the input or independent variable approaches positive or negative infinity. You also need to consider the sign of the coefficient of the variable under the radical sign. By following these steps and avoiding common mistakes, you can accurately determine the end behavior of a radical function.

References

• [1] "Calculus" by Michael Spivak
• [2] "Algebra" by Michael Artin
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

If you want to learn more about the end behavior of functions, I recommend checking out the following resources:

• Khan Academy: End Behavior of Functions
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: End Behavior of Functions