Which Best Describes The Range Of The Function $f(x)=2\left(\frac{1}{4}\right)^x$ After It Has Been Reflected Over The $y$-axis?A. All Real Numbers B. All Real Numbers Less Than 0 C. All Real Numbers Greater Than 0 D. All Real

Introduction

When dealing with functions, reflecting them over the y-axis is a common operation that can help us understand their behavior and properties. In this article, we will explore the concept of reflecting a function over the y-axis and how it affects the range of the function. We will use the function $f(x)=2\left(\frac{1}{4}\right)^x$ as a case study to illustrate this concept.

What is a Reflection Over the Y-Axis?

A reflection over the y-axis is a transformation that flips a function over the y-axis, effectively changing the sign of the x-coordinate of each point on the graph. In other words, if we have a function $f(x)$, its reflection over the y-axis is given by $f(-x)$.

How Does Reflection Affect the Range of a Function?

When a function is reflected over the y-axis, its range is also affected. The range of a function is the set of all possible output values it can produce for a given input. In the case of a reflection over the y-axis, the range of the function is also reflected, but with a twist.

The Function $f(x)=2\left(\frac{1}{4}\right)^x$

Let's consider the function $f(x)=2\left(\frac{1}{4}\right)^x$. This function is an exponential function with a base of $\frac{1}{4}$ and a coefficient of 2. The graph of this function is a curve that approaches the x-axis as x approaches negative infinity and approaches the y-axis as x approaches positive infinity.

The Reflection of $f(x)=2\left(\frac{1}{4}\right)^x$ Over the Y-Axis

To reflect the function $f(x)=2\left(\frac{1}{4}\right)^x$ over the y-axis, we need to replace x with -x in the function. This gives us the reflected function $f(-x)=2\left(\frac{1}{4}\right)^{-x}$.

Simplifying the Reflected Function

We can simplify the reflected function $f(-x)=2\left(\frac{1}{4}\right)^{-x}$ by using the property of exponents that states $a^{-x}=\frac{1}{a^x}$. Applying this property to the reflected function, we get:

$f(-x)=2\left(\frac{1}{4}\right)^{-x}=2\left(\frac{4}{1}\right)^x=2\cdot4^x$

The Range of the Reflected Function

Now that we have simplified the reflected function, we can determine its range. The range of a function is the set of all possible output values it can produce for a given input. In the case of the reflected function $f(-x)=2\cdot4^x$, the range is all real numbers greater than 0.

Conclusion

In conclusion, reflecting a function over the y-axis can affect its range in a significant way. In the case of the function $f(x)=2\left(\frac{1}{4}\right)^x$, its reflection over the y-axis results in a function with a range of all real numbers greater than 0. This is because the reflection of the function changes the sign of the x-coordinate of each point on the graph, effectively flipping the function over the y-axis.

Answer

Based on our analysis, the correct answer is:

C. All real numbers greater than 0

Final Thoughts

In this article, we explored the concept of reflecting a function over the y-axis and how it affects the range of the function. We used the function $f(x)=2\left(\frac{1}{4}\right)^x$ as a case study to illustrate this concept. By reflecting the function over the y-axis, we were able to determine its range and understand how the reflection affects the function's behavior.

References

• [1] "Functions and Graphs" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Algebra and Trigonometry" by Michael Sullivan

Additional Resources

• Khan Academy: Reflections of Functions
• Mathway: Reflections of Functions
• Wolfram Alpha: Reflections of Functions
  Q&A: Reflections of Functions ==============================

Introduction

In our previous article, we explored the concept of reflecting a function over the y-axis and how it affects the range of the function. We used the function $f(x)=2\left(\frac{1}{4}\right)^x$ as a case study to illustrate this concept. In this article, we will answer some frequently asked questions about reflections of functions.

Q: What is a reflection over the y-axis?

A: A reflection over the y-axis is a transformation that flips a function over the y-axis, effectively changing the sign of the x-coordinate of each point on the graph. In other words, if we have a function $f(x)$, its reflection over the y-axis is given by $f(-x)$.

Q: How does reflection affect the range of a function?

A: When a function is reflected over the y-axis, its range is also affected. The range of a function is the set of all possible output values it can produce for a given input. In the case of a reflection over the y-axis, the range of the function is also reflected, but with a twist.

Q: What is the range of the function $f(x)=2\left(\frac{1}{4}\right)^x$ after it has been reflected over the y-axis?

A: The range of the function $f(x)=2\left(\frac{1}{4}\right)^x$ after it has been reflected over the y-axis is all real numbers greater than 0.

Q: How do I determine the range of a reflected function?

A: To determine the range of a reflected function, you need to replace x with -x in the original function and then simplify the resulting expression. This will give you the reflected function, and you can then determine its range.

Q: Can I reflect a function over the x-axis?

A: Yes, you can reflect a function over the x-axis. To do this, you need to replace y with -y in the original function. This will give you the reflected function, and you can then determine its range.

Q: What is the difference between reflecting a function over the y-axis and reflecting it over the x-axis?

A: Reflecting a function over the y-axis changes the sign of the x-coordinate of each point on the graph, while reflecting it over the x-axis changes the sign of the y-coordinate of each point on the graph.

Q: Can I reflect a function over both the x-axis and the y-axis?

A: Yes, you can reflect a function over both the x-axis and the y-axis. To do this, you need to replace x with -x and y with -y in the original function. This will give you the reflected function, and you can then determine its range.

Q: How do I graph a reflected function?

A: To graph a reflected function, you need to graph the original function and then reflect it over the y-axis or x-axis. You can use a graphing calculator or software to help you graph the function.

Conclusion

In conclusion, reflecting a function over the y-axis or x-axis can affect its range in a significant way. By understanding how reflection affects the range of a function, you can better analyze and graph functions. We hope this Q&A article has helped you understand the concept of reflections of functions.

Additional Resources

• Khan Academy: Reflections of Functions
• Mathway: Reflections of Functions
• Wolfram Alpha: Reflections of Functions