The Function F ( X ) = 1 6 ( 2 5 ) X F(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x F ( X ) = 6 1 ​ ( 5 2 ​ ) X Is Reflected Across The Y Y Y -axis To Create The Function G ( X G(x G ( X ]. Which Ordered Pair Is On G ( X G(x G ( X ]?A. { (-3, \frac{4}{375})$} B . \[ B. \[ B . \[ (-2,

Introduction

In mathematics, reflecting a function across the y-axis is a fundamental concept that helps us understand the properties of functions and their transformations. When a function is reflected across the y-axis, its x-values are negated, resulting in a new function that is symmetric to the original function with respect to the y-axis. In this article, we will explore the concept of reflecting a function across the y-axis and apply it to the given function $f(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x$ to find the ordered pair on the reflected function $g(x)$.

Understanding Function Reflection

To reflect a function across the y-axis, we need to negate the x-values of the function. This means that for a function $f(x)$, the reflected function $g(x)$ is given by $g(x) = f(-x)$. In other words, we replace $x$ with $-x$ in the original function to get the reflected function.

Applying Function Reflection to the Given Function

The given function is $f(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x$. To reflect this function across the y-axis, we need to negate the x-values. This means that we replace $x$ with $-x$ in the original function to get the reflected function $g(x)$.

$g(x) = f(-x) = \frac{1}{6}\left(\frac{2}{5}\right)^{-x}$

Simplifying the Reflected Function

To simplify the reflected function, we can use the property of exponents that states $\left(\frac{a}{b}\right)^{-x} = \left(\frac{b}{a}\right)^x$. Applying this property to the reflected function, we get:

$g(x) = \frac{1}{6}\left(\frac{2}{5}\right)^{-x} = \frac{1}{6}\left(\frac{5}{2}\right)^x$

Finding the Ordered Pair on the Reflected Function

To find the ordered pair on the reflected function $g(x)$, we need to find the value of $g(-3)$. Substituting $x = -3$ into the reflected function, we get:

$g(-3) = \frac{1}{6}\left(\frac{5}{2}\right)^{-3}$

Evaluating the Expression

To evaluate the expression, we can use the property of exponents that states $\left(\frac{a}{b}\right)^{-x} = \left(\frac{b}{a}\right)^x$. Applying this property to the expression, we get:

$g(-3) = \frac{1}{6}\left(\frac{5}{2}\right)^{-3} = \frac{1}{6}\left(\frac{2}{5}\right)^3$

Simplifying the Expression

To simplify the expression, we can evaluate the exponent:

$g(-3) = \frac{1}{6}\left(\frac{2}{5}\right)^3 = \frac{1}{6} \cdot \frac{8}{125} = \frac{4}{375}$

Conclusion

In conclusion, the ordered pair on the reflected function $g(x)$ is $(-3, \frac{4}{375})$. This means that the point $(-3, \frac{4}{375})$ lies on the reflected function $g(x)$.

Answer

The correct answer is:

A. $(-3, \frac{4}{375})$

Discussion

The concept of reflecting a function across the y-axis is a fundamental concept in mathematics that helps us understand the properties of functions and their transformations. In this article, we applied this concept to the given function $f(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x$ to find the ordered pair on the reflected function $g(x)$. We simplified the reflected function and evaluated the expression to find the ordered pair.

References

• [1] "Function Reflection Across the Y-Axis". Math Open Reference.
• [2] "Exponents and Powers". Khan Academy.

Related Topics

• Function Reflection Across the X-Axis
• Function Translation
• Function Scaling
  Q&A: Function Reflection Across the Y-Axis =============================================

Introduction

In our previous article, we explored the concept of reflecting a function across the y-axis and applied it to the given function $f(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x$ to find the ordered pair on the reflected function $g(x)$. In this article, we will answer some frequently asked questions related to function reflection across the y-axis.

Q: What is function reflection across the y-axis?

A: Function reflection across the y-axis is a process of reflecting a function across the y-axis, which means that the x-values of the function are negated. This results in a new function that is symmetric to the original function with respect to the y-axis.

Q: How do I reflect a function across the y-axis?

A: To reflect a function across the y-axis, you need to negate the x-values of the function. This means that for a function $f(x)$, the reflected function $g(x)$ is given by $g(x) = f(-x)$.

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

A: Reflecting a function across the x-axis involves negating the y-values of the function, while reflecting a function across the y-axis involves negating the x-values of the function.

Q: Can I reflect a function across the y-axis multiple times?

A: Yes, you can reflect a function across the y-axis multiple times. Each time you reflect the function, the x-values are negated, resulting in a new function that is symmetric to the original function with respect to the y-axis.

Q: How do I find the ordered pair on a reflected function?

A: To find the ordered pair on a reflected function, you need to substitute the x-value into the reflected function and evaluate the expression.

Q: What is the formula for reflecting a function across the y-axis?

A: The formula for reflecting a function across the y-axis is $g(x) = f(-x)$.

Q: Can I use function reflection across the y-axis to solve problems in real-world applications?

A: Yes, function reflection across the y-axis can be used to solve problems in real-world applications, such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when reflecting a function across the y-axis?

A: Some common mistakes to avoid when reflecting a function across the y-axis include:

• Negating the y-values instead of the x-values
• Not using the correct formula for reflecting a function across the y-axis
• Not evaluating the expression correctly

Conclusion

In conclusion, function reflection across the y-axis is a fundamental concept in mathematics that helps us understand the properties of functions and their transformations. By answering some frequently asked questions related to function reflection across the y-axis, we hope to provide a better understanding of this concept and its applications.

References

• [1] "Function Reflection Across the Y-Axis". Math Open Reference.
• [2] "Exponents and Powers". Khan Academy.

Related Topics

• Function Reflection Across the X-Axis
• Function Translation
• Function Scaling