How are the graphs of the functions [tex]$f(x)=\sqrt{16}^x$[/tex] and [tex]$g(x)=\sqrt[3]{64}^x$[/tex] related?
A. The functions [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] are equivalent.
B. The function [tex]$g(x)$[/tex] increases at a faster rate.
C. The function [tex]$g(x)$[/tex] has a greater initial value.
D. The function [tex]$g(x)$[/tex] decreases at a faster rate.
Answer (2)
Simplify f ( x ) = 16 x to 4 x .
Simplify g ( x ) = 3 64 x to 4 x .
Compare the simplified forms and determine that f ( x ) = g ( x ) .
Conclude that the functions are equivalent: The functions f ( x ) and g ( x ) are equivalent.
Explanation
Understanding the Problem We are given two functions, f ( x ) = 16 x and g ( x ) = 3 64 x , and we want to determine how their graphs are related. The possible relationships are that the functions are equivalent, that g ( x ) increases at a faster rate, that g ( x ) has a greater initial value, or that g ( x ) decreases at a faster rate.
Simplifying f(x) First, let's simplify the expression for f ( x ) . We have f ( x ) = 16 x . Since 16 = 4 , we can rewrite f ( x ) as f ( x ) = 4 x .
Simplifying g(x) Next, let's simplify the expression for g ( x ) . We have g ( x ) = 3 64 x . Since 3 64 = 4 , we can rewrite g ( x ) as g ( x ) = 4 x .
Comparing the Functions Now, let's compare the simplified forms of f ( x ) and g ( x ) . We found that f ( x ) = 4 x and g ( x ) = 4 x . Since the two functions are equal, their graphs are identical, meaning the functions are equivalent.
Final Answer Therefore, the graphs of the functions f ( x ) = 16 x and g ( x ) = 3 64 x are equivalent. The correct answer is: The functions f ( x ) and g ( x ) are equivalent.
Examples
Imagine you're comparing the growth of two investments. One investment grows at a rate of 16 per year, and the other grows at a rate of 3 64 per year. By simplifying these rates, you find that both investments actually grow at the same rate of 4 per year. This means that over time, the two investments will perform identically, showing the practical importance of understanding equivalent exponential functions.
The functions f ( x ) and g ( x ) simplify to the same expression, 4 x , showing that they are equivalent. Thus, their graphs are identical. The correct choice is: The functions f ( x ) and g ( x ) are equivalent.
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