The Function F ( X ) = X 3 F(x)=\sqrt[3]{x} F ( X ) = 3 X ​ Is Reflected Over The X X X -axis To Create The Graph Of G ( X ) = − X 3 G(x)=-\sqrt[3]{x} G ( X ) = − 3 X ​ . Which Is The Graph Of G ( X G(x G ( X ]?

Introduction

In mathematics, the concept of reflecting a function over the x-axis is a fundamental idea in graphing and algebra. When a function is reflected over the x-axis, its graph is flipped upside down, resulting in a new function. In this article, we will explore the reflection of the function $f(x)=\sqrt[3]{x}$ over the x-axis to create the graph of $g(x)=-\sqrt[3]{x}$.

Understanding the Original Function

The original function $f(x)=\sqrt[3]{x}$ is a cubic root function, which means that it takes the cube root of the input value $x$. This function is defined for all real numbers $x$, and its graph is a curve that increases as $x$ increases. The graph of $f(x)$ is a simple, smooth curve that passes through the origin $(0,0)$.

Reflection Over the x-axis

When a function is reflected over the x-axis, its graph is flipped upside down. This means that the y-coordinate of each point on the graph is negated. In other words, if the original function has a point $(x,y)$ on its graph, the reflected function will have the point $(x,-y)$ on its graph.

The Reflected Function

The reflected function $g(x)=-\sqrt[3]{x}$ is obtained by negating the y-coordinate of each point on the graph of $f(x)$. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but flipped upside down. The graph of $g(x)$ is also a cubic root function, but with a negative sign in front of the cube root.

Graphing the Reflected Function

To graph the reflected function $g(x)=-\sqrt[3]{x}$, we can start by graphing the original function $f(x)=\sqrt[3]{x}$. Then, we can flip the graph upside down to obtain the graph of $g(x)$. This can be done by negating the y-coordinate of each point on the graph of $f(x)$.

Key Features of the Reflected Function

The graph of the reflected function $g(x)=-\sqrt[3]{x}$ has several key features. First, it is a cubic root function, just like the original function $f(x)$. Second, it is defined for all real numbers $x$, just like the original function. Third, its graph is a curve that increases as $x$ increases, just like the graph of the original function.

Comparison with the Original Function

The graph of the reflected function $g(x)=-\sqrt[3]{x}$ is similar to the graph of the original function $f(x)=\sqrt[3]{x}$, but with a negative sign in front of the cube root. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but flipped upside down.

Conclusion

In conclusion, the function $f(x)=\sqrt[3]{x}$ is reflected over the x-axis to create the graph of $g(x)=-\sqrt[3]{x}$. The graph of $g(x)$ is the same as the graph of $f(x)$, but flipped upside down. This means that the graph of $g(x)$ is a cubic root function, defined for all real numbers $x$, and with a negative sign in front of the cube root.

Final Answer

The graph of the reflected function $g(x)=-\sqrt[3]{x}$ is the same as the graph of the original function $f(x)=\sqrt[3]{x}$, but flipped upside down. This means that the graph of $g(x)$ is a cubic root function, defined for all real numbers $x$, and with a negative sign in front of the cube root.

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Algebra and Trigonometry" by Michael Sullivan
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Additional Resources

• [1] Khan Academy: Functions and Graphs
• [2] MIT OpenCourseWare: Algebra and Trigonometry
• [3] Wolfram Alpha: Cubic Root Function
  

Introduction

In our previous article, we explored the reflection of the function $f(x)=\sqrt[3]{x}$ over the x-axis to create the graph of $g(x)=-\sqrt[3]{x}$. In this article, we will answer some common questions related to this topic.

Q&A

Q: What is the difference between the original function $f(x)=\sqrt[3]{x}$ and the reflected function $g(x)=-\sqrt[3]{x}$?

A: The main difference between the two functions is that the reflected function $g(x)$ has a negative sign in front of the cube root, while the original function $f(x)$ does not. This means that the graph of $g(x)$ is the same as the graph of $f(x)$, but flipped upside down.

Q: How do I graph the reflected function $g(x)=-\sqrt[3]{x}$?

A: To graph the reflected function $g(x)$, you can start by graphing the original function $f(x)=\sqrt[3]{x}$. Then, you can flip the graph upside down to obtain the graph of $g(x)$. This can be done by negating the y-coordinate of each point on the graph of $f(x)$.

Q: What are some key features of the reflected function $g(x)=-\sqrt[3]{x}$?

A: Some key features of the reflected function $g(x)$ include:

• It is a cubic root function, just like the original function $f(x)$.
• It is defined for all real numbers $x$, just like the original function.
• Its graph is a curve that increases as $x$ increases, just like the graph of the original function.

Q: How does the reflected function $g(x)=-\sqrt[3]{x}$ compare to the original function $f(x)=\sqrt[3]{x}$?

A: The graph of the reflected function $g(x)$ is the same as the graph of the original function $f(x)$, but flipped upside down. This means that the graph of $g(x)$ is a cubic root function, defined for all real numbers $x$, and with a negative sign in front of the cube root.

Q: What are some real-world applications of the reflected function $g(x)=-\sqrt[3]{x}$?

A: The reflected function $g(x)$ has several real-world applications, including:

• Modeling population growth and decline.
• Analyzing financial data and predicting stock prices.
• Studying the behavior of complex systems and predicting their outcomes.

Q: How can I use the reflected function $g(x)=-\sqrt[3]{x}$ in my own work or research?

A: The reflected function $g(x)$ can be used in a variety of ways, including:

• Modeling real-world phenomena and predicting their outcomes.
• Analyzing data and identifying trends and patterns.
• Developing new mathematical models and theories.

Conclusion

In conclusion, the reflected function $g(x)=-\sqrt[3]{x}$ is a powerful tool for modeling and analyzing real-world phenomena. By understanding the properties and behavior of this function, you can gain insights into complex systems and make predictions about their outcomes.

Final Answer

The reflected function $g(x)=-\sqrt[3]{x}$ is a cubic root function, defined for all real numbers $x$, and with a negative sign in front of the cube root. Its graph is the same as the graph of the original function $f(x)=\sqrt[3]{x}$, but flipped upside down.

References

• [1] "Functions and Graphs" by Michael Sullivan
• [2] "Algebra and Trigonometry" by Michael Sullivan
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Additional Resources

• [1] Khan Academy: Functions and Graphs
• [2] MIT OpenCourseWare: Algebra and Trigonometry
• [3] Wolfram Alpha: Cubic Root Function