Select The Correct Answer.What Is The End Behavior Of This Radical Function?$f(x) = 4 \sqrt{x - 6}$A. As $x$ Approaches Positive Infinity, $f(x$\] Approaches Positive Infinity. B. As $x$ Approaches Negative Infinity,

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Introduction

Radical functions, also known as root functions, are a type of mathematical function that involves a variable under a radical sign. In this article, we will explore the end behavior of the radical function f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6}, which is a specific type of radical function. We will examine the behavior of this function as xx approaches positive and negative infinity.

What is End Behavior?

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

The Radical Function f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6}

The given radical function is f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6}. To understand the end behavior of this function, we need to analyze its behavior as xx approaches positive and negative infinity.

As xx Approaches Positive Infinity

As xx approaches positive infinity, the value of xβˆ’6x - 6 also approaches positive infinity. Since the square root function is an increasing function, the value of xβˆ’6\sqrt{x - 6} will also approach positive infinity. Therefore, as xx approaches positive infinity, f(x)f(x) approaches positive infinity.

As xx Approaches Negative Infinity

As xx approaches negative infinity, the value of xβˆ’6x - 6 also approaches negative infinity. Since the square root function is an increasing function, the value of xβˆ’6\sqrt{x - 6} will approach zero as xx approaches negative infinity. Therefore, as xx approaches negative infinity, f(x)f(x) approaches zero.

Conclusion

In conclusion, the end behavior of the radical function f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6} is as follows:

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

Answer

Based on our analysis, the correct answer is:

A. As xx approaches positive infinity, f(x)f(x) approaches positive infinity.

Discussion

The end behavior of a radical function depends on the behavior of the function inside the radical sign. In this case, the function inside the radical sign is xβˆ’6x - 6, which approaches positive infinity as xx approaches positive infinity and approaches negative infinity as xx approaches negative infinity. Therefore, the end behavior of the radical function f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6} is determined by the behavior of the function inside the radical sign.

Example Problems

Here are some example problems that illustrate the end behavior of radical functions:

  • Find the end behavior of the radical function f(x)=3x+2f(x) = 3 \sqrt{x + 2}.
  • Find the end behavior of the radical function f(x)=2xβˆ’1f(x) = 2 \sqrt{x - 1}.

Solutions

  • The end behavior of the radical function f(x)=3x+2f(x) = 3 \sqrt{x + 2} is as follows:
  • As xx approaches positive infinity, f(x)f(x) approaches positive infinity.
  • As xx approaches negative infinity, f(x)f(x) approaches zero.
  • The end behavior of the radical function f(x)=2xβˆ’1f(x) = 2 \sqrt{x - 1} is as follows:
  • As xx approaches positive infinity, f(x)f(x) approaches positive infinity.
  • As xx approaches negative infinity, f(x)f(x) approaches zero.

Key Takeaways

In conclusion, the end behavior of a radical function depends on the behavior of the function inside the radical sign. By analyzing the behavior of the function inside the radical sign, we can determine the end behavior of the radical function. The key takeaways from this article are:

  • The end behavior of a radical function depends on the behavior of the function inside the radical sign.
  • As xx approaches positive infinity, the value of the function inside the radical sign approaches positive infinity, and the value of the radical function approaches positive infinity.
  • As xx approaches negative infinity, the value of the function inside the radical sign approaches negative infinity, and the value of the radical function approaches zero.

References

Introduction

Radical functions, also known as root functions, are a type of mathematical function that involves a variable under a radical sign. In this article, we will explore the end behavior of the radical function f(x)=4xβˆ’6f(x) = 4 \sqrt{x - 6}, which is a specific type of radical function. We will also answer some frequently asked questions about radical functions.

Q&A

Q: What is a radical function?

A: A radical function is a type of mathematical function that involves a variable under a radical sign. It is also known as a root function.

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

A: The end behavior of a radical function depends on the behavior of the function inside the radical sign. If the function inside the radical sign approaches positive infinity, the value of the radical function will also approach positive infinity. If the function inside the radical sign approaches negative infinity, the value of the radical function will approach zero.

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 function inside the radical sign. If the function inside the radical sign is a linear function, you can use the following rules:

  • If the coefficient of the linear function is positive, the value of the radical function will approach positive infinity as the input variable approaches positive infinity.
  • If the coefficient of the linear function is negative, the value of the radical function will approach zero as the input variable approaches negative infinity.

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

A: A radical function is a type of mathematical function that involves a variable under a radical sign, while a polynomial function is a type of mathematical function that involves a variable raised to a power. Radical functions and polynomial functions have different properties and behaviors.

Q: Can I use the same rules to determine the end behavior of a polynomial function as I do for a radical function?

A: No, you cannot use the same rules to determine the end behavior of a polynomial function as you do for a radical function. Polynomial functions have different properties and behaviors than radical functions, and you need to use different rules to determine their end behavior.

Q: How do I graph a radical function?

A: To graph a radical function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph of the function.

Q: Can I use a radical function to model real-world phenomena?

A: Yes, you can use a radical function to model real-world phenomena. Radical functions can be used to model situations where the relationship between the input and output variables is not linear.

Example Problems

Here are some example problems that illustrate the end behavior of radical functions:

  • Find the end behavior of the radical function f(x)=3x+2f(x) = 3 \sqrt{x + 2}.
  • Find the end behavior of the radical function f(x)=2xβˆ’1f(x) = 2 \sqrt{x - 1}.

Solutions

  • The end behavior of the radical function f(x)=3x+2f(x) = 3 \sqrt{x + 2} is as follows:
  • As xx approaches positive infinity, f(x)f(x) approaches positive infinity.
  • As xx approaches negative infinity, f(x)f(x) approaches zero.
  • The end behavior of the radical function f(x)=2xβˆ’1f(x) = 2 \sqrt{x - 1} is as follows:
  • As xx approaches positive infinity, f(x)f(x) approaches positive infinity.
  • As xx approaches negative infinity, f(x)f(x) approaches zero.

Key Takeaways

In conclusion, radical functions are a type of mathematical function that involves a variable under a radical sign. The end behavior of a radical function depends on the behavior of the function inside the radical sign. By analyzing the behavior of the function inside the radical sign, you can determine the end behavior of the radical function. The key takeaways from this article are:

  • The end behavior of a radical function depends on the behavior of the function inside the radical sign.
  • As xx approaches positive infinity, the value of the function inside the radical sign approaches positive infinity, and the value of the radical function approaches positive infinity.
  • As xx approaches negative infinity, the value of the function inside the radical sign approaches negative infinity, and the value of the radical function approaches zero.

References