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

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Introduction

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

Understanding the Function

The function $f(x)=-2 \sqrt[3]{x+7}$ involves a cube root, which means that the value inside the radical sign is cubed. The negative sign in front of the function indicates that the function is a decreasing function. As $x$ increases, the value inside the radical sign increases, and the cube root of the value increases. However, the negative sign in front of the function causes the function to decrease as $x$ increases.

End Behavior of the Function

To determine the end behavior of the function, we need to consider what happens as $x$ approaches positive or negative infinity. As $x$ approaches positive infinity, the value inside the radical sign, $x+7$, also approaches positive infinity. Since the cube root of a positive number is also positive, the function $f(x)=-2 \sqrt[3]{x+7}$ approaches negative infinity as $x$ approaches positive infinity.

On the other hand, as $x$ approaches negative infinity, the value inside the radical sign, $x+7$, approaches negative infinity. However, the cube root of a negative number is also negative, and the negative sign in front of the function causes the function to approach positive infinity as $x$ approaches 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 positive infinity, $f(x)$ approaches negative infinity.
• As $x$ approaches negative infinity, $f(x)$ approaches positive infinity.

Therefore, the correct answer is:

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

However, this is incorrect. The correct answer is:

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

This is because the function $f(x)=-2 \sqrt[3]{x+7}$ approaches negative infinity as $x$ approaches positive infinity, not negative infinity.

Example

To illustrate this concept, let's consider an example. Suppose we want to find the end behavior of the function $f(x)=-2 \sqrt[3]{x+7}$ as $x$ approaches positive infinity.

We can start by plugging in a large positive value for $x$, such as $x=1000$. This gives us:

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

Since $1007$ is a large positive number, the cube root of $1007$ is also a large positive number. Therefore, the function $f(1000)=-2 \sqrt[3]{1007}$ is a large negative number.

As we continue to increase the value of $x$, the value inside the radical sign, $x+7$, also increases. This causes the cube root of the value to increase, and the function $f(x)=-2 \sqrt[3]{x+7}$ to approach negative infinity.

Graphical Representation

To visualize the end behavior of the function, we can graph the function on a coordinate plane. The graph of the function $f(x)=-2 \sqrt[3]{x+7}$ is a decreasing function that approaches negative infinity as $x$ approaches positive infinity.

The graph of the function can be represented as follows:

• As $x$ approaches positive infinity, the graph of the function approaches the horizontal asymptote $y=-\infty$.
• As $x$ approaches negative infinity, the graph of the function approaches the horizontal asymptote $y=\infty$.

Key Takeaways

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

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

Therefore, the correct answer is:

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

This is because the function $f(x)=-2 \sqrt[3]{x+7}$ approaches negative infinity as $x$ approaches positive infinity, not negative infinity.

Final Answer

The final answer is:

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

Introduction

In our previous article, we explored the end behavior of the radical function $f(x)=-2 \sqrt[3]{x+7}$. In this article, we will answer some common questions related to 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 $x$ approaches positive or negative infinity. It describes what happens to the function as $x$ gets very large or very small.

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

A: To determine the end behavior of a radical function, you need to consider the sign of the coefficient in front of the radical and the sign of the value inside the radical. If the coefficient is negative, the function will approach negative infinity as $x$ approaches positive infinity. If the coefficient is positive, the function will approach positive infinity as $x$ approaches negative infinity.

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 will approach a specific value as $x$ approaches positive or negative infinity, while a radical function will approach positive or negative infinity.

Q: Can a radical function have a horizontal asymptote?

A: Yes, a radical function can have a horizontal asymptote. However, the horizontal asymptote will be at positive or negative infinity, depending on the sign of the coefficient in front of the radical.

Q: How do you graph a radical function?

A: To graph a radical function, you need to start by finding the domain of the function. Then, you can use a table of values to find the corresponding $y$-values. Finally, you can plot the points on a coordinate plane and draw a smooth curve through the points.

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 understand the behavior of the function as $x$ approaches positive or negative infinity. This is important in many real-world applications, such as physics and engineering.

Q: Can a radical function have a vertical asymptote?

A: Yes, a radical function can have a vertical asymptote. A vertical asymptote occurs when the value inside the radical is equal to zero.

Q: How do you determine the vertical asymptote of a radical function?

A: To determine the vertical asymptote of a radical function, you need to set the value inside the radical equal to zero and solve for $x$. This will give you the value of $x$ where the vertical asymptote occurs.

Q: What is the difference between a vertical asymptote and a horizontal asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches as $x$ approaches a specific value. A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches positive or negative infinity.

Q: Can a radical function have both a vertical and a horizontal asymptote?

A: Yes, a radical function can have both a vertical and a horizontal asymptote. However, the vertical asymptote will occur at a specific value of $x$, while the horizontal asymptote will occur at positive or negative infinity.

Conclusion

In conclusion, the end behavior of a radical function is an important concept in mathematics. It helps us understand the behavior of the function as $x$ approaches positive or negative infinity. By understanding the end behavior of a radical function, we can better understand the behavior of the function in different situations.

Final Answer

The final answer is:

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

This is because the function $f(x)=-2 \sqrt[3]{x+7}$ approaches negative infinity as $x$ approaches positive infinity, not negative infinity.