To determine the function [tex]\( g(x) \)[/tex] that represents the reflection of [tex]\( f(x) = 6 \left(\frac{1}{3}\right)^x \)[/tex] across the [tex]\( y \)[/tex]-axis, we need to follow these steps:
1. Understand the given function:
The given function is [tex]\( f(x) = 6 \left(\frac{1}{3}\right)^x \)[/tex].
2. Reflect [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis:
When we reflect a function across the [tex]\( y \)[/tex]-axis, we replace [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the function. Therefore, we have:
[tex]\[
f(-x) = 6 \left(\frac{1}{3}\right)^{-x}
\][/tex]
3. Simplify the expression:
Recall the property of exponents that [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex] and apply it to our function:
[tex]\[
\left( \frac{1}{3} \right)^{-x} = 3^x
\][/tex]
Therefore, replacing [tex]\( x \)[/tex] with [tex]\( -x \)[/tex] in the original function [tex]\( f(x) \)[/tex], the new function becomes:
[tex]\[
f(-x) = 6 \cdot 3^x
\][/tex]
Thus, the reflection of [tex]\( f(x) = 6 \left( \frac{1}{3} \right)^x \)[/tex] across the [tex]\( y \)[/tex]-axis is:
[tex]\[
g(x) = 6 \cdot 3^x
\][/tex]
Consequently, the correct choice is:
\[
g(x) = 6(3)^x
\
