Question

which function represents g(x), a reflection of f(x) = \frac{1}{2}(3)^x across the y-axis?\
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$g(x) = 2(3)^{-x}$\
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$g(x) = -\frac{1}{2}(3)^x$\
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$g(x) = \frac{1}{2}(3)^{-x}$\
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$g(x) = 2(3)^x$

Explanation:

Step1: Recall reflection rule

To reflect a function \( f(x) \) across the \( y \)-axis, we replace \( x \) with \( -x \) in the function. So the rule is \( g(x)=f(-x) \).

Step2: Apply the rule to \( f(x) \)

Given \( f(x)=\frac{1}{2}(3)^{x} \), substitute \( x \) with \( -x \):
\( g(x)=f(-x)=\frac{1}{2}(3)^{-x} \).

Answer:

\( g(x)=\frac{1}{2}(3)^{-x} \) (the third option)