The Function $f(x)=5\left(\frac{1}{5}\right)^x$ Is Reflected Over The $y$-axis. Which Equations Represent The Reflected Function? Select Two Options.A. $f(x)=\frac{1}{5}(5)^{-x}$B. $f(x)=\frac{1}{5}

Reflection is a fundamental concept in mathematics that involves flipping a function over a specific axis. In this article, we will explore the concept of reflection and apply it to the given function $f(x)=5\left(\frac{1}{5}\right)^x$. We will analyze the function and determine which equations represent the reflected function.

Understanding the Given Function

The given function is $f(x)=5\left(\frac{1}{5}\right)^x$. This function represents an exponential function with a base of $\frac{1}{5}$ and a coefficient of $5$. The function is defined for all real values of $x$.

Reflection Over the $y$-Axis

When a function is reflected over the $y$-axis, the $x$-coordinates of the function are negated. In other words, if the original function is $f(x)$, the reflected function is $f(-x)$. To reflect the given function over the $y$-axis, we need to replace $x$ with $-x$ in the original function.

Determining the Reflected Function

To determine the reflected function, we substitute $-x$ for $x$ in the original function:

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

Using the property of exponents that states $\left(\frac{1}{a}\right)^{-x}=a^x$, we can rewrite the function as:

$f(-x)=5\left(5\right)^{-x}$

This is the reflected function of the original function $f(x)=5\left(\frac{1}{5}\right)^x$.

Comparing the Reflected Function with the Options

Now that we have determined the reflected function, we can compare it with the options provided:

A. $f(x)=\frac{1}{5}(5)^{-x}$

B. $f(x)=\frac{1}{5}(5)^x$

The reflected function we determined is $f(-x)=5\left(5\right)^{-x}$. This can be rewritten as $f(-x)=\frac{1}{5}(5)^{-x}$, which matches option A.

Conclusion

In this article, we explored the concept of reflection in mathematics and applied it to the given function $f(x)=5\left(\frac{1}{5}\right)^x$. We determined the reflected function by replacing $x$ with $-x$ in the original function and compared it with the options provided. The reflected function is represented by option A: $f(x)=\frac{1}{5}(5)^{-x}$.

Key Takeaways

• Reflection is a fundamental concept in mathematics that involves flipping a function over a specific axis.
• To reflect a function over the $y$-axis, we need to replace $x$ with $-x$ in the original function.
• The reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$ is $f(-x)=\frac{1}{5}(5)^{-x}$.

Frequently Asked Questions

• What is reflection in mathematics? Reflection in mathematics is the process of flipping a function over a specific axis.
• How do we reflect a function over the $y$-axis? To reflect a function over the $y$-axis, we need to replace $x$ with $-x$ in the original function.
• What is the reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$? The reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$ is $f(-x)=\frac{1}{5}(5)^{-x}$.
  Frequently Asked Questions: Reflection in Mathematics =====================================================

In our previous article, we explored the concept of reflection in mathematics and applied it to the given function $f(x)=5\left(\frac{1}{5}\right)^x$. We determined the reflected function by replacing $x$ with $-x$ in the original function and compared it with the options provided. In this article, we will answer some frequently asked questions related to reflection in mathematics.

Q&A: Reflection in Mathematics

Q: What is reflection in mathematics?

A: Reflection in mathematics is the process of flipping a function over a specific axis. It involves replacing the variable $x$ with $-x$ in the original function to obtain the reflected function.

Q: How do we reflect a function over the $y$-axis?

A: To reflect a function over the $y$-axis, we need to replace $x$ with $-x$ in the original function. This will result in the reflected function, which is obtained by negating the $x$-coordinates of the original function.

Q: What is the reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$?

A: The reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$ is $f(-x)=\frac{1}{5}(5)^{-x}$. This can be obtained by replacing $x$ with $-x$ in the original function.

Q: How do we reflect a function over the $x$-axis?

A: To reflect a function over the $x$-axis, we need to multiply the original function by $-1$. This will result in the reflected function, which is obtained by negating the $y$-coordinates of the original function.

Q: What is the reflected function of $f(x)=x^2$ over the $x$-axis?

A: The reflected function of $f(x)=x^2$ over the $x$-axis is $f(x)=-x^2$. This can be obtained by multiplying the original function by $-1$.

Q: How do we reflect a function over the line $y=x$?

A: To reflect a function over the line $y=x$, we need to interchange the $x$ and $y$ variables in the original function. This will result in the reflected function, which is obtained by swapping the $x$ and $y$ coordinates of the original function.

Q: What is the reflected function of $f(x)=x^2$ over the line $y=x$?

A: The reflected function of $f(x)=x^2$ over the line $y=x$ is $f(x)=x^2$. This can be obtained by interchanging the $x$ and $y$ variables in the original function.

Conclusion

In this article, we answered some frequently asked questions related to reflection in mathematics. We discussed the concept of reflection, how to reflect a function over the $y$-axis, $x$-axis, and the line $y=x$, and provided examples to illustrate the concepts. We hope that this article has provided a clear understanding of reflection in mathematics and has helped to clarify any doubts that readers may have had.

Key Takeaways

• Reflection in mathematics is the process of flipping a function over a specific axis.
• To reflect a function over the $y$-axis, we need to replace $x$ with $-x$ in the original function.
• To reflect a function over the $x$-axis, we need to multiply the original function by $-1$.
• To reflect a function over the line $y=x$, we need to interchange the $x$ and $y$ variables in the original function.

Frequently Asked Questions (FAQs)

• What is reflection in mathematics? Reflection in mathematics is the process of flipping a function over a specific axis.
• How do we reflect a function over the $y$-axis? To reflect a function over the $y$-axis, we need to replace $x$ with $-x$ in the original function.
• What is the reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$? The reflected function of $f(x)=5\left(\frac{1}{5}\right)^x$ is $f(-x)=\frac{1}{5}(5)^{-x}$.
• How do we reflect a function over the $x$-axis? To reflect a function over the $x$-axis, we need to multiply the original function by $-1$.
• What is the reflected function of $f(x)=x^2$ over the $x$-axis? The reflected function of $f(x)=x^2$ over the $x$-axis is $f(x)=-x^2$.
• How do we reflect a function over the line $y=x$? To reflect a function over the line $y=x$, we need to interchange the $x$ and $y$ variables in the original function.
• What is the reflected function of $f(x)=x^2$ over the line $y=x$? The reflected function of $f(x)=x^2$ over the line $y=x$ is $f(x)=x^2$.