To determine which function represents [tex]\( g(x) \)[/tex], the reflection of [tex]\( f(x) = 6 \left( \frac{1}{3} \right)^x \)[/tex] across the [tex]\( y \)[/tex]-axis, we need to understand the effect of reflecting a function across the [tex]\( y \)[/tex]-axis.
When a function [tex]\( f(x) \)[/tex] is reflected across the [tex]\( y \)[/tex]-axis, the new function [tex]\( g(x) \)[/tex] is given by [tex]\( f(-x) \)[/tex]. Therefore, we need to evaluate [tex]\( f(-x) \)[/tex] for [tex]\( f(x) = 6 \left( \frac{1}{3} \right)^x \)[/tex].
1. Start with the original function:
[tex]\[
f(x) = 6 \left( \frac{1}{3} \right)^x
\][/tex]
2. Reflect [tex]\( f(x) \)[/tex] across the [tex]\( y \)[/tex]-axis by substituting [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(x) = f(-x) = 6 \left( \frac{1}{3} \right)^{-x}
\][/tex]
3. Simplify the expression [tex]\( 6 \left( \frac{1}{3} \right)^{-x} \)[/tex].
Recall that [tex]\( \left( \frac{1}{3} \right)^{-x} \)[/tex] can be rewritten using properties of exponents. Specifically, [tex]\( \left( \frac{1}{3} \right)^{-x} = 3^x \)[/tex].
Thus,
[tex]\[
g(x) = 6 (3)^x
\][/tex]
4. Compare with the given options:
- [tex]\( g(x) = -6 \left( \frac{1}{3} \right)^x \)[/tex]
- [tex]\( g(x) = -6 \left( \frac{1}{3} \right)^{-x} \)[/tex]
- [tex]\( g(x) = 6 (3)^x \)[/tex]
- [tex]\( g(x) = 6 (3)^{-x} \)[/tex]
The function [tex]\( g(x) = 6 (3)^x \)[/tex] matches our result from reflecting the original function across the [tex]\( y \)[/tex]-axis.
Therefore, the correct function is:
[tex]\[
\boxed{g(x) = 6 (3)^x}
\][/tex] which corresponds to the third option.
Hence, the function that represents [tex]\( g(x) \)[/tex] is:
[tex]\[
\boxed{3}
\][/tex]
