The Function $g(x$\] Is A Transformation Of The Cube Root Parent Function, $f(x)=\sqrt[3]{x}$. What Function Is $g(x$\]?A. $g(x)=\sqrt[3]{x+2}$ B. $g(x)=2 \sqrt[3]{x}$ C. $g(x)=\frac{1}{2}

Introduction

In mathematics, transformations of functions are essential concepts that help us understand how functions can be manipulated and represented in different ways. The cube root parent function, $f(x)=\sqrt[3]{x}$, is a fundamental function that can be transformed in various ways to create new functions. In this article, we will explore the transformation of the cube root parent function to obtain the function $g(x)$.

Understanding the Cube Root Parent Function

The cube root parent function, $f(x)=\sqrt[3]{x}$, is a function that takes a real number $x$ as input and returns its cube root as output. This function is a basic example of a root function, which is a function that takes a real number as input and returns a real number as output. The cube root function is a special case of a root function, where the index of the root is 3.

Transformations of the Cube Root Parent Function

Transformations of the cube root parent function involve changing the input or output of the function in some way. There are several types of transformations that can be applied to the cube root parent function, including:

• Vertical Stretching: This involves multiplying the output of the function by a constant factor.
• Vertical Compressing: This involves dividing the output of the function by a constant factor.
• Horizontal Stretching: This involves replacing $x$ with $ax$, where $a$ is a constant factor.
• Horizontal Compressing: This involves replacing $x$ with $\frac{x}{a}$, where $a$ is a constant factor.
• Shifting: This involves adding or subtracting a constant value from the input or output of the function.

Finding the Function $g(x)$

To find the function $g(x)$, we need to apply a transformation to the cube root parent function. Let's consider the options given:

A. $g(x)=\sqrt[3]{x+2}$ B. $g(x)=2 \sqrt[3]{x}$ C. $g(x)=\frac{1}{2} \sqrt[3]{x}$

We can analyze each option to determine which one represents the function $g(x)$.

Option A: $g(x)=\sqrt[3]{x+2}$

This option represents a horizontal shift of the cube root parent function by 2 units to the left. This means that the input of the function is shifted by 2 units to the left, resulting in a new function that takes the input $x+2$ and returns its cube root as output.

Option B: $g(x)=2 \sqrt[3]{x}$

This option represents a vertical stretching of the cube root parent function by a factor of 2. This means that the output of the function is multiplied by 2, resulting in a new function that takes the input $x$ and returns 2 times its cube root as output.

Option C: $g(x)=\frac{1}{2} \sqrt[3]{x}$

This option represents a vertical compressing of the cube root parent function by a factor of $\frac{1}{2}$. This means that the output of the function is divided by 2, resulting in a new function that takes the input $x$ and returns $\frac{1}{2}$ times its cube root as output.

Conclusion

Based on our analysis, we can conclude that the function $g(x)$ is represented by option B: $g(x)=2 \sqrt[3]{x}$. This function is a vertical stretching of the cube root parent function by a factor of 2, resulting in a new function that takes the input $x$ and returns 2 times its cube root as output.

Final Answer

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

Introduction

In our previous article, we explored the transformation of the cube root parent function to obtain the function $g(x)$. We analyzed three options and concluded that the function $g(x)$ is represented by option B: $g(x)=2 \sqrt[3]{x}$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information about the function $g(x)$.

Q&A

Q: What is the cube root parent function?

A: The cube root parent function is a function that takes a real number $x$ as input and returns its cube root as output. It is represented by the equation $f(x)=\sqrt[3]{x}$.

Q: What is the function $g(x)$?

A: The function $g(x)$ is a transformation of the cube root parent function. It is represented by the equation $g(x)=2 \sqrt[3]{x}$.

Q: What type of transformation is represented by the function $g(x)$?

A: The function $g(x)$ represents a vertical stretching of the cube root parent function by a factor of 2.

Q: What is the effect of the vertical stretching on the output of the function?

A: The vertical stretching multiplies the output of the function by 2, resulting in a new function that takes the input $x$ and returns 2 times its cube root as output.

Q: Can the function $g(x)$ be represented by any of the other options?

A: No, the function $g(x)$ cannot be represented by options A or C. Option A represents a horizontal shift of the cube root parent function, while option C represents a vertical compressing of the cube root parent function.

Q: What is the significance of the function $g(x)$ in mathematics?

A: The function $g(x)$ is an example of a transformation of a parent function, which is an essential concept in mathematics. Understanding transformations of functions is crucial in algebra, geometry, and other areas of mathematics.

Q: Can the function $g(x)$ be used in real-world applications?

A: Yes, the function $g(x)$ can be used in real-world applications, such as modeling population growth, chemical reactions, and other phenomena that involve cube root relationships.

Conclusion

In this Q&A article, we provided additional information and clarification about the function $g(x)$, which is a transformation of the cube root parent function. We hope that this article has helped to answer any questions and provide a better understanding of the function $g(x)$.

Final Answer

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