The Function $f(x)=\frac{1}{6}\left(\frac{2}{5}\right)^x$ Is Reflected Across The $y$-axis To Create The Function $g(x$\]. Which Ordered Pair Is On $g(x$\]?A. $\left(-3, \frac{4}{375}\right$\]B. $\left(-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 behavior. When a function is reflected across the y-axis, its x-values are negated, resulting in a new function. 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 if we have a function $f(x)$, the reflected function $g(x)$ will have the x-values negated, i.e., $g(x) = f(-x)$. This concept is crucial in understanding the behavior of functions and their properties.

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, resulting in the function $g(x) = f(-x)$. Substituting $-x$ into the function, we get:

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

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:

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

Finding the Ordered Pair

To find the ordered pair on the reflected function $g(x)$, we need to find the value of $x$ that satisfies the equation $g(x) = \frac{4}{375}$. Substituting the value of $g(x)$, we get:

$$\frac{1}{6}\left(\frac{5}{2}\right)^x = \frac{4}{375}$$

To solve for $x$, we can start by isolating the exponential term:

$$\left(\frac{5}{2}\right)^x = \frac{4}{375} \times 6$$

Simplifying the right-hand side, we get:

$$\left(\frac{5}{2}\right)^x = \frac{24}{375}$$

To solve for $x$, we can take the logarithm of both sides. Using the logarithmic identity $\log_a b^c = c \log_a b$, we get:

$$x \log_{\frac{5}{2}} \frac{24}{375} = \log_{\frac{5}{2}} \frac{24}{375}$$

Simplifying the logarithmic expression, we get:

$$x = \log_{\frac{5}{2}} \frac{24}{375}$$

Using a calculator to evaluate the logarithmic expression, we get:

$$x \approx -3$$

Therefore, the ordered pair on the reflected function $g(x)$ is $\left(-3, \frac{4}{375}\right)$.

Conclusion

In this 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 $g(x)$. We simplified the reflected function and used logarithmic properties to solve for the value of $x$ that satisfies the equation $g(x) = \frac{4}{375}$. The ordered pair on the reflected function $g(x)$ is $\left(-3, \frac{4}{375}\right)$.

Answer

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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 provide a Q&A guide to help you understand the concept of function reflection and its applications.

Q: What is function reflection?

A: Function reflection is the process of negating the x-values of a function to create a new function. This means that if we have a function $f(x)$, the reflected function $g(x)$ will have the x-values negated, i.e., $g(x) = f(-x)$.

Q: Why is function reflection important?

A: Function reflection is an important concept in mathematics because it helps us understand the properties of functions and their behavior. By reflecting a function across the y-axis, we can gain insights into the function's symmetry, periodicity, and other properties.

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 if you have a function $f(x)$, the reflected function $g(x)$ will have the x-values negated, i.e., $g(x) = f(-x)$.

Q: What is the difference between a function and its reflection?

A: The main difference between a function and its reflection is the sign of the x-values. If you have a function $f(x)$, its reflection $g(x)$ will have the x-values negated, i.e., $g(x) = f(-x)$. This means that the graph of the reflected function will be a mirror image of the original function across the y-axis.

Q: Can I reflect a function more than once?

A: Yes, you can reflect a function more than once. Each time you reflect a function, you will get a new function with the x-values negated. For example, if you have a function $f(x)$ and you reflect it twice, you will get a function $g(x)$ that is the reflection of the original function.

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 find the value of $x$ that satisfies the equation $g(x) = y$. This means that you need to solve for $x$ in the equation $g(x) = y$, where $g(x)$ is the reflected function.

Q: What is the relationship between the original function and its reflection?

A: The relationship between the original function and its reflection is that the reflection is a mirror image of the original function across the y-axis. This means that the graph of the reflected function will be a mirror image of the original function.

Q: Can I use function reflection to solve problems?

A: Yes, you can use function reflection to solve problems. By reflecting a function across the y-axis, you can gain insights into the function's properties and behavior, which can help you solve problems.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of function reflection and its applications. We covered topics such as what function reflection is, why it is important, how to reflect a function, and how to find the ordered pair on a reflected function. We hope that this guide has been helpful in understanding the concept of function reflection.

Answer Key

1. What is function reflection?
   • Function reflection is the process of negating the x-values of a function to create a new function.
2. Why is function reflection important?
   • Function reflection is an important concept in mathematics because it helps us understand the properties of functions and their behavior.
3. How do I reflect a function across the y-axis?
   • To reflect a function across the y-axis, you need to negate the x-values of the function.
4. What is the difference between a function and its reflection?
   • The main difference between a function and its reflection is the sign of the x-values.
5. Can I reflect a function more than once?
   • Yes, you can reflect a function more than once.
6. How do I find the ordered pair on a reflected function?
   • To find the ordered pair on a reflected function, you need to find the value of $x$ that satisfies the equation $g(x) = y$.
7. What is the relationship between the original function and its reflection?
   • The relationship between the original function and its reflection is that the reflection is a mirror image of the original function across the y-axis.
8. Can I use function reflection to solve problems?
   • Yes, you can use function reflection to solve problems.