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 \( x \).

The functions \( f(x) = 16^x \) and \( g(x) = 16^{\frac{1}{2} x} \) are actually closely related through the properties of exponents. In fact, \( g(x) \) can be rewritten as \( g(x) = \sqrt{16^x} \) because raising a number to the one-half power is the same as taking the square root. This highlights that every output of \( g(x) \) is the square root of the corresponding output of \( f(x) \). In other words, when you input a value \( x \), \( g(x) \) outputs precisely the square root of what \( f(x) \) outputs. So, for any value of \( x \), if you calculate both \( f(x) \) and \( g(x) \), you’ll find that \( g(x) = \sqrt{f(x)} \). This is a great example of how exponent rules can simplify understanding the relationship between different exponential functions!