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

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Introduction

Radical functions, also known as root functions, are a type of mathematical function that involves a variable or expression under a root, such as a square root or cube root. These functions can be used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits. In this article, we will explore the end behavior of radical functions, specifically the function f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7}.

What is End Behavior?

End behavior refers to the behavior of a function as the input or independent variable approaches positive or negative infinity. In other words, it describes what happens to the function as the input gets very large or very small. Understanding the end behavior of a function is crucial in mathematics, as it helps us predict the behavior of the function in different scenarios.

The Function f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7}

The given function is f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7}. This function involves a cube root, which is a type of radical function. 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+7x+7 is taken and multiplied by −2-2.

As xx Approaches Positive Infinity

To determine the end behavior of the function as xx approaches positive infinity, we need to evaluate the limit of the function as xx gets very large. We can do this by substituting xx with a large value, such as 10001000 or 1000010000, and evaluating the function.

f(1000) = -2 \sqrt[3]{1000+7} = -2 \sqrt[3]{1007} ≈ -2(9.98) ≈ -19.96

As we can see, the function approaches negative infinity as xx approaches positive infinity. This is because the cube root of a large number is also large, and multiplying it by −2-2 gives a large negative value.

As xx Approaches Negative Infinity

To determine the end behavior of the function as xx approaches negative infinity, we need to evaluate the limit of the function as xx gets very small. We can do this by substituting xx with a small value, such as −1000-1000 or −10000-10000, and evaluating the function.

f(-1000) = -2 \sqrt[3]{-1000+7} = -2 \sqrt[3]{-993} ≈ -2(-9.98) ≈ 19.96

As we can see, the function approaches positive infinity as xx approaches negative infinity. This is because the cube root of a negative number is also negative, and multiplying it by −2-2 gives a positive value.

Conclusion

In conclusion, the end behavior of the radical function f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7} is as follows:

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

This is because the cube root of a large number is also large, and multiplying it by −2-2 gives a large negative value. Similarly, the cube root of a negative number is also negative, and multiplying it by −2-2 gives a positive value.

Key Takeaways

  • The end behavior of a function describes what happens to the function as the input gets very large or very small.
  • Radical functions, such as the cube root function, can be used to model real-world phenomena.
  • The end behavior of a radical function depends on the sign of the input and the coefficient of the radical.

Final Thoughts

Introduction

In our previous article, we explored the end behavior of the radical function f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7}. We discussed how the function approaches negative infinity as xx approaches positive infinity and approaches positive infinity as xx approaches negative infinity. 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 describes what happens to the function as the input or independent variable approaches positive or negative infinity. It describes the behavior of the function 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 evaluate the limit of the function as the input approaches positive or negative infinity. You can do this by substituting the input with a large value, such as 1000 or 10000, and evaluating the function.

Q: What happens to the end behavior of a radical function if the coefficient is negative?

A: If the coefficient of the radical function is negative, the end behavior of the function will be opposite of what it would be if the coefficient were positive. For example, if the function is f(x)=−2x+73f(x) = -2 \sqrt[3]{x+7}, the end behavior will be opposite of what it would be if the function were f(x)=2x+73f(x) = 2 \sqrt[3]{x+7}.

Q: Can the end behavior of a radical function be determined by looking at the graph of the function?

A: Yes, the end behavior of a radical function can be determined by looking at the graph of the function. If the graph of the function approaches negative infinity as xx approaches positive infinity, the end behavior of the function is negative infinity. If the graph of the function approaches positive infinity as xx approaches negative infinity, the end behavior of the function is positive infinity.

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

A: To determine the end behavior of a radical function with a negative exponent, you need to evaluate the limit of the function as the input approaches positive or negative infinity. You can do this by substituting the input with a large value, such as 1000 or 10000, and evaluating the function.

Q: Can the end behavior of a radical function be determined by looking at the equation of the function?

A: Yes, the end behavior of a radical function can be determined by looking at the equation of the function. If the equation of the function involves a cube root, the end behavior of the function will be determined by the sign of the input and the coefficient of the cube root.

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

A: The end behavior of a radical function is significant because it helps us predict the behavior of the function in different scenarios. By understanding the end behavior of a function, we can gain insights into its behavior and make predictions about its behavior in different situations.

Conclusion

In conclusion, the end behavior of a radical function is an important concept in mathematics that helps us predict the behavior of the function in different scenarios. By understanding the end behavior of a function, we can gain insights into its behavior and make predictions about its behavior in different situations. We hope that this article has helped you understand the end behavior of radical functions and has provided you with a better understanding of this important concept.

Key Takeaways

  • The end behavior of a radical function describes what happens to the function as the input or independent variable approaches positive or negative infinity.
  • The end behavior of a radical function can be determined by evaluating the limit of the function as the input approaches positive or negative infinity.
  • The end behavior of a radical function can be determined by looking at the graph of the function or the equation of the function.
  • The end behavior of a radical function is significant because it helps us predict the behavior of the function in different scenarios.

Final Thoughts

Understanding the end behavior of radical functions is crucial in mathematics, as it helps us predict the behavior of the function in different scenarios. By analyzing the end behavior of a function, we can gain insights into its behavior and make predictions about its behavior in different situations. We hope that this article has helped you understand the end behavior of radical functions and has provided you with a better understanding of this important concept.