How are the functions $$ f(x)=16^{x} $$ and $$ g(x)=16^{\frac{1}{2} x} $$ related? The output values of $$ g(x) $$ are one-half the output values of $$ f(x) $$ for the same value of $$ x $$. The output values of $$ g(x) $$ are one-quarter the output values of $$ f(x) $$ for the same value of $$ x $$. The output values of $$ g(x) $$ are the square root of the output values of $$ f(x) $$ for the same value of $$ x $$. The output values of $$ g(x) $$ are the square of the output values of $$ f(x) $$ for the same value of $$ x $$.

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To determine the relationship between the functions $$ f(x) = 16^{x} $$ and $$ g(x) = 16^{\frac{1}{2}x} $$, let's analyze $$ g(x) $$ in terms of $$ f(x) $$.

1. **Express $$ g(x) $$ in terms of $$ f(x) $$:**
   $$
   g(x) = 16^{\frac{1}{2}x} = (16^{x})^{\frac{1}{2}} = \sqrt{16^{x}} = \sqrt{f(x)}
   $$
   
   This shows that $$ g(x) $$ is the square root of $$ f(x) $$.

2. **Interpretation:**
   - For any given value of $$ x $$, the output of $$ g(x) $$ is the square root of the output of $$ f(x) $$.
   - Mathematically, if $$ f(x) = y $$, then $$ g(x) = \sqrt{y} $$.

Therefore, the correct relationship is:

**The output values of $$ g(x) $$ are the square root of the output values of $$ f(x) $$ for the same value of $$ x $$.**

**Answer:**  
The output values of $$ g(x) $$ are the square root of the output values of $$ f(x) $$ for the same value of $$ x $$.
The output values of $$ g(x) $$ are the square root of the output values of $$ f(x) $$ for the same value of $$ x $$.

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