How are the functions $f(x)=16^x$ and $g(x)=16^{\frac{1}{2} x}$ related?
A. The output values of $g(x)$ are one-half the output values of $f(x)$ for the same value of $x$.
B. The output values of $g(x)$ are one-quarter the output values of $f(x)$ for the same value of $x$.
C. The output values of $g(x)$ are the square root of the output values of $f(x)$ for the same value of $x$.
D. The output values of $g(x)$ are the square of the output values of $f(x)$ for the same value of $x$.
Answer (2)
Express g ( x ) in terms of f ( x ) : g ( x ) = 1 6 2 1 x = ( 1 6 x ) 2 1 = ( f ( x ) ) 2 1 .
This simplifies to g ( x ) = f ( x ) .
Verify with an example: f ( 2 ) = 1 6 2 = 256 and g ( 2 ) = 1 6 2 1 ⋅ 2 = 16 , and 256 = 16 .
Therefore, the output values of g ( x ) are the square root of the output values of f ( x ) for the same value of x , so the answer is The output values of g ( x ) are the square root of the output values of f ( x ) for the same value of x . .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 1 6 x and g ( x ) = 1 6 2 1 x , and we need to determine the relationship between their output values for the same value of x . The options suggest relationships involving fractions, square roots, or squares.
Expressing g(x) in terms of f(x) Let's express g ( x ) in terms of f ( x ) . We can rewrite g ( x ) as follows: g ( x ) = 1 6 2 1 x = ( 1 6 x ) 2 1 Since f ( x ) = 1 6 x , we can substitute f ( x ) into the expression for g ( x ) :
g ( x ) = ( f ( x ) ) 2 1 This means that g ( x ) is the square root of f ( x ) .
Verification with an Example Now, let's verify this relationship with an example. Let's take x = 2 . Then: f ( 2 ) = 1 6 2 = 256 g ( 2 ) = 1 6 2 1 ⋅ 2 = 1 6 1 = 16 Now, let's check if g ( 2 ) is the square root of f ( 2 ) :
f ( 2 ) = 256 = 16 Since g ( 2 ) = 16 and f ( 2 ) = 16 , the relationship g ( x ) = f ( x ) holds true.
Identifying the Correct Relationship Comparing the derived relationship with the given options, we find that 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 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the relationship between exponential functions, like f ( x ) = 1 6 x and g ( x ) = 1 6 2 1 x , can help in analyzing and predicting these phenomena. For example, if f ( x ) represents the population size at time x , then g ( x ) could represent a related quantity, such as the rate of resource consumption, which grows at a rate proportional to the square root of the population size.
The functions f ( x ) = 1 6 x and g ( x ) = 1 6 2 1 x are related as g ( x ) is the square root of f ( x ) . Therefore, the correct option is C. 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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