Select The Correct Answer From Each Drop-down Menu.The Function F ( X ) = X 3 F(x) = X^3 F ( X ) = X 3 Has Been Transformed, Resulting In Function M M M : M ( X ) = 1 3 X 3 + 6 M(x) = \frac{1}{3} X^3 + 6 M ( X ) = 3 1 ​ X 3 + 6 As X X X Approaches Positive Infinity, M ( X M(x M ( X ]

Introduction

In mathematics, function transformations are essential concepts that help us understand how functions change under various operations. These transformations can be vertical, horizontal, or a combination of both. In this article, we will focus on understanding how a function $f(x) = x^3$ is transformed into a new function $m(x) = \frac{1}{3} x^3 + 6$. We will analyze the behavior of the transformed function as $x$ approaches positive infinity.

Function Transformations

A function transformation is a process of changing a function $f(x)$ into a new function $g(x)$ by applying a set of operations. These operations can be vertical shifts, horizontal shifts, stretches, or compressions. In the case of the function $m(x) = \frac{1}{3} x^3 + 6$, we can see that it is a transformation of the original function $f(x) = x^3$.

Vertical Transformation

The function $m(x) = \frac{1}{3} x^3 + 6$ is a vertical transformation of the original function $f(x) = x^3$. This is because the coefficient of $x^3$ has been changed from 1 to $\frac{1}{3}$. This vertical transformation affects the amplitude of the function, making it narrower or wider.

Horizontal Transformation

However, there is no horizontal transformation in this case, as the coefficient of $x^3$ is not affected by a horizontal shift. The function $m(x) = \frac{1}{3} x^3 + 6$ is a vertical transformation of the original function $f(x) = x^3$.

Behavior as $x$ Approaches Positive Infinity

As $x$ approaches positive infinity, the function $m(x) = \frac{1}{3} x^3 + 6$ behaves in a specific way. Since the coefficient of $x^3$ is $\frac{1}{3}$, the function will approach positive infinity as $x$ approaches positive infinity. This is because the term $\frac{1}{3} x^3$ will dominate the function as $x$ becomes very large.

Analysis of the Function

To analyze the function $m(x) = \frac{1}{3} x^3 + 6$, we can use the concept of limits. As $x$ approaches positive infinity, the limit of the function $m(x)$ is:

$$\lim_{x\to\infty} m(x) = \lim_{x\to\infty} \frac{1}{3} x^3 + 6$$

Using the properties of limits, we can simplify this expression:

$$\lim_{x\to\infty} m(x) = \lim_{x\to\infty} \frac{1}{3} x^3 + 6 = \infty$$

This means that as $x$ approaches positive infinity, the function $m(x) = \frac{1}{3} x^3 + 6$ approaches positive infinity.

Conclusion

In conclusion, the function $m(x) = \frac{1}{3} x^3 + 6$ is a vertical transformation of the original function $f(x) = x^3$. As $x$ approaches positive infinity, the function $m(x)$ approaches positive infinity. This is because the term $\frac{1}{3} x^3$ dominates the function as $x$ becomes very large.

Select the Correct Answer

Based on the analysis above, we can conclude that as $x$ approaches positive infinity, the function $m(x) = \frac{1}{3} x^3 + 6$ approaches:

• Positive Infinity
• Negative Infinity
• Zero
• A Constant Value

The correct answer is Positive Infinity.

Final Answer

Introduction

In our previous article, we discussed the concept of function transformations in mathematics and analyzed the behavior of a transformed function $m(x) = \frac{1}{3} x^3 + 6$ as $x$ approaches positive infinity. In this article, we will answer some frequently asked questions related to function transformations and provide additional insights into this important topic.

Q: What is a function transformation?

A: A function transformation is a process of changing a function $f(x)$ into a new function $g(x)$ by applying a set of operations. These operations can be vertical shifts, horizontal shifts, stretches, or compressions.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

• Vertical transformations: These involve changing the amplitude of the function, making it narrower or wider.
• Horizontal transformations: These involve changing the period of the function, making it longer or shorter.
• Stretches and compressions: These involve changing the scale of the function, making it larger or smaller.

Q: How do I determine the type of function transformation?

A: To determine the type of function transformation, you need to analyze the function and identify the operations that have been applied. For example, if the function has been shifted vertically, you can identify the type of transformation by looking at the coefficient of the $x^3$ term.

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

A: A vertical transformation involves changing the amplitude of the function, while a horizontal transformation involves changing the period of the function. For example, if the function $f(x) = x^3$ is transformed into $m(x) = \frac{1}{3} x^3 + 6$, the vertical transformation has changed the amplitude of the function, while the horizontal transformation has not changed the period.

Q: How do I analyze the behavior of a transformed function?

A: To analyze the behavior of a transformed function, you need to use the concept of limits. For example, if the function $m(x) = \frac{1}{3} x^3 + 6$ is analyzed as $x$ approaches positive infinity, the limit of the function can be used to determine its behavior.

Q: What is the significance of function transformations in mathematics?

A: Function transformations are an essential concept in mathematics, as they help us understand how functions change under various operations. This knowledge is crucial in many areas of mathematics, including calculus, algebra, and geometry.

Q: How do I apply function transformations in real-world problems?

A: Function transformations can be applied in many real-world problems, such as modeling population growth, analyzing economic data, and understanding the behavior of physical systems. By applying function transformations, you can gain insights into the behavior of complex systems and make informed decisions.

Conclusion

In conclusion, function transformations are an essential concept in mathematics that helps us understand how functions change under various operations. By analyzing the behavior of transformed functions, we can gain insights into the behavior of complex systems and make informed decisions. We hope that this Q&A article has provided you with a better understanding of function transformations and their significance in mathematics.

Final Answer

The final answer is: $\boxed{\infty}$