The Function $f(x$\] Is Defined Below. What Is The End Behavior Of $f(x$\]?$f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$Answer:A. As $x \rightarrow \infty, F(x) \rightarrow \infty$ And As $x

Introduction

In mathematics, the end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity. Understanding the end behavior of a function is crucial in various mathematical and real-world applications. In this article, we will explore the end behavior of the function $f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$.

The Function $f(x)$

The function $f(x)$ is a polynomial function of degree 6, which means it has six terms with the highest power of $x$ being 6. The function can be written as:

$$f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$$

End Behavior of $f(x)$

To determine the end behavior of $f(x)$, we need to analyze the leading term of the function, which is the term with the highest power of $x$. In this case, the leading term is $4x^6$. Since the exponent of the leading term is positive (6), we can conclude that the end behavior of $f(x)$ will be determined by the term $4x^6$.

As $x \rightarrow \infty, f(x) \rightarrow \infty$

As $x$ approaches positive infinity, the term $4x^6$ will dominate the function $f(x)$. Since the exponent of the leading term is positive (6), the function will increase without bound as $x$ approaches positive infinity. Therefore, we can conclude that as $x \rightarrow \infty, f(x) \rightarrow \infty$.

As $x \rightarrow -\infty, f(x) \rightarrow -\infty$

As $x$ approaches negative infinity, the term $4x^6$ will still dominate the function $f(x)$. However, since the exponent of the leading term is positive (6), the function will decrease without bound as $x$ approaches negative infinity. Therefore, we can conclude that as $x \rightarrow -\infty, f(x) \rightarrow -\infty$.

Conclusion

In conclusion, the end behavior of the function $f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$ is determined by the leading term $4x^6$. As $x$ approaches positive infinity, the function will increase without bound, and as $x$ approaches negative infinity, the function will decrease without bound.

Key Takeaways

• The end behavior of a function is determined by the leading term of the function.
• If the exponent of the leading term is positive, the function will increase without bound as $x$ approaches positive infinity.
• If the exponent of the leading term is positive, the function will decrease without bound as $x$ approaches negative infinity.

Real-World Applications

Understanding the end behavior of a function is crucial in various real-world applications, such as:

• Physics: The end behavior of a function can be used to model the behavior of physical systems, such as the motion of objects under the influence of gravity.
• Engineering: The end behavior of a function can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: The end behavior of a function can be used to model the behavior of economic systems, such as the behavior of supply and demand curves.

Final Thoughts

Introduction

In our previous article, we explored the end behavior of the function $f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$. We determined that the end behavior of the function is determined by the leading term $4x^6$. As $x$ approaches positive infinity, the function will increase without bound, and as $x$ approaches negative infinity, the function will decrease without bound. In this article, we will answer some frequently asked questions about the end behavior of the function $f(x)$.

Q&A

Q: What is the end behavior of a function?

A: The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity.

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

A: To determine the end behavior of a function, you need to analyze the leading term of the function, which is the term with the highest power of $x$.

Q: What is the leading term of the function $f(x)$?

A: The leading term of the function $f(x)$ is $4x^6$.

Q: As $x$ approaches positive infinity, what happens to the function $f(x)$?

A: As $x$ approaches positive infinity, the function $f(x)$ will increase without bound.

Q: As $x$ approaches negative infinity, what happens to the function $f(x)$?

A: As $x$ approaches negative infinity, the function $f(x)$ will decrease without bound.

Q: Can the end behavior of a function be determined by other terms in the function?

A: No, the end behavior of a function can only be determined by the leading term of the function.

Q: What is the significance of the exponent of the leading term in determining the end behavior of a function?

A: The exponent of the leading term determines whether the function will increase or decrease without bound as $x$ approaches positive or negative infinity.

Q: Can the end behavior of a function be used to model real-world phenomena?

A: Yes, the end behavior of a function can be used to model real-world phenomena, such as the behavior of physical systems, economic systems, and more.

Real-World Applications

Understanding the end behavior of a function is crucial in various real-world applications, such as:

• Physics: The end behavior of a function can be used to model the behavior of physical systems, such as the motion of objects under the influence of gravity.
• Engineering: The end behavior of a function can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: The end behavior of a function can be used to model the behavior of economic systems, such as the behavior of supply and demand curves.

Conclusion

In conclusion, the end behavior of the function $f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$ is determined by the leading term $4x^6$. As $x$ approaches positive infinity, the function will increase without bound, and as $x$ approaches negative infinity, the function will decrease without bound. Understanding the end behavior of a function is crucial in various mathematical and real-world applications.

Key Takeaways

• The end behavior of a function is determined by the leading term of the function.
• The exponent of the leading term determines whether the function will increase or decrease without bound as $x$ approaches positive or negative infinity.
• Understanding the end behavior of a function is crucial in various mathematical and real-world applications.

Final Thoughts

In conclusion, the end behavior of the function $f(x) = 120x^4 - 2112x + 44x^5 - 1152 + 4x^6 - 1280x^2 - 160x^3$ is determined by the leading term $4x^6$. As $x$ approaches positive infinity, the function will increase without bound, and as $x$ approaches negative infinity, the function will decrease without bound. Understanding the end behavior of a function is crucial in various mathematical and real-world applications.