Which Function Represents { G(x) $}$, A Reflection Of { F(x) = 4 \left(\frac{1}{2}\right)^x $}$ Across The { X $}$-axis?A. { G(x) = -4(2)^x $}$B. { G(x) = 4(2)^{-x} $} C . \[ C. \[ C . \[ G(x) =
Reflection Across the x-axis: Understanding the Concept and Identifying the Correct Function
In mathematics, a reflection across the x-axis is a fundamental concept that involves flipping a function over the x-axis. This operation changes the sign of the function's output, resulting in a new function that is a mirror image of the original function. In this article, we will explore the concept of reflection across the x-axis and identify the correct function that represents a reflection of a given function, { f(x) = 4 \left(\frac{1}{2}\right)^x $}$, across the x-axis.
A reflection across the x-axis is a transformation that flips a function over the x-axis. This operation changes the sign of the function's output, resulting in a new function that is a mirror image of the original function. To reflect a function across the x-axis, we multiply the function by -1.
Mathematical Representation of Reflection Across the x-axis
The mathematical representation of a reflection across the x-axis can be expressed as:
{ g(x) = -f(x) $}$
where { g(x) $}$ is the reflected function and { f(x) $}$ is the original function.
Reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ Across the x-axis
To reflect the function { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis, we multiply the function by -1.
{ g(x) = -f(x) = -4 \left(\frac{1}{2}\right)^x $}$
However, we can simplify the expression by using the property of exponents that states { \left(\frac{1}{2}\right)^x = \left(2{-1}\right)x = 2^{-x} $}$.
{ g(x) = -4 \left(2{-1}\right)x = -4 \left(2^{-x}\right) $}$
Now that we have identified the correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis, let's compare the options.
A. { g(x) = -4(2)^x $}$
B. { g(x) = 4(2)^{-x} $}$
C. { g(x) = -4(2)^{-x} $}$
The correct option is C. { g(x) = -4(2)^{-x} $}$, which is equivalent to our derived function.
In conclusion, a reflection across the x-axis is a fundamental concept in mathematics that involves flipping a function over the x-axis. To reflect a function across the x-axis, we multiply the function by -1. In this article, we identified the correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis and compared the options. The correct option is C. { g(x) = -4(2)^{-x} $}$, which is equivalent to our derived function.
The final answer is C. { g(x) = -4(2)^{-x} $}$.
Reflection Across the x-axis: Q&A
In our previous article, we explored the concept of reflection across the x-axis and identified the correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis. In this article, we will answer some frequently asked questions related to reflection across the x-axis.
Q: What is reflection across the x-axis?
A: Reflection across the x-axis is a transformation that flips a function over the x-axis. This operation changes the sign of the function's output, resulting in a new function that is a mirror image of the original function.
Q: How do I reflect a function across the x-axis?
A: To reflect a function across the x-axis, you multiply the function by -1.
Q: What is the mathematical representation of reflection across the x-axis?
A: The mathematical representation of a reflection across the x-axis can be expressed as:
{ g(x) = -f(x) $}$
where { g(x) $}$ is the reflected function and { f(x) $}$ is the original function.
Q: How do I simplify the expression of a reflected function?
A: To simplify the expression of a reflected function, you can use the property of exponents that states { \left(\frac{1}{2}\right)^x = \left(2{-1}\right)x = 2^{-x} $}$.
Q: What is the correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis?
A: The correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis is:
{ g(x) = -4 \left(2{-1}\right)x = -4 \left(2^{-x}\right) $}$
Q: How do I compare the options for a reflected function?
A: To compare the options for a reflected function, you can substitute the original function into the options and simplify the expressions.
Q: What is the final answer for the reflected function of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis?
A: The final answer for the reflected function of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis is:
{ g(x) = -4(2)^{-x} $}$
In conclusion, reflection across the x-axis is a fundamental concept in mathematics that involves flipping a function over the x-axis. In this article, we answered some frequently asked questions related to reflection across the x-axis and provided the correct function that represents a reflection of { f(x) = 4 \left(\frac{1}{2}\right)^x $}$ across the x-axis.
The final answer is { g(x) = -4(2)^{-x} $}$.