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 conclude that f ( x ) and g ( x ) are equivalent.
The functions f ( x ) and g ( x ) are equivalent, meaning their graphs are identical. The functions f ( x ) and g ( x ) are equivalent.
Explanation
Analyze the problem We are given two functions, f ( x ) = 16 x and g ( x ) = 3 64 x , and we need to determine how their graphs are related. The options are:
The functions f ( x ) and g ( x ) are equivalent.
The function g ( x ) increases at a faster rate.
The function g ( x ) has a greater initial value.
The function g ( x ) decreases at a faster rate.
Simplify f(x) To determine the relationship between the graphs, we need to simplify the expressions for f ( x ) and g ( x ) . Let's start with f ( x ) . We can rewrite 16 as 1 6 2 1 . So, we have
f ( x ) = 16 x = ( 1 6 2 1 ) x
Further simplification of f(x) Since 16 = 4 2 , we can further simplify f ( x ) as follows:
f ( x ) = ( 1 6 2 1 ) x = (( 4 2 ) 2 1 ) x = ( 4 2 ⋅ 2 1 ) x = 4 x
Simplify g(x) Now, let's simplify g ( x ) . We can rewrite 3 64 as 6 4 3 1 . So, we have
g ( x ) = 3 64 x = ( 6 4 3 1 ) x
Further simplification of g(x) Since 64 = 4 3 , we can further simplify g ( x ) as follows:
g ( x ) = ( 6 4 3 1 ) x = (( 4 3 ) 3 1 ) x = ( 4 3 ⋅ 3 1 ) x = 4 x
Compare f(x) and g(x) Comparing the simplified forms of f ( x ) and g ( x ) , we see that f ( x ) = 4 x and g ( x ) = 4 x . Therefore, the two functions are equivalent.
Conclusion Since f ( x ) = g ( x ) = 4 x , the graphs of the two functions are identical. This means that the functions are equivalent.
Examples
Exponential functions like these are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the population size can be modeled by an exponential function. Similarly, the amount of money in a bank account with compound interest grows exponentially over time. Understanding the relationship between different exponential functions can help us make predictions and analyze these phenomena more effectively.
The functions f ( x ) and g ( x ) are equivalent because both simplify to 4 x . Therefore, their graphs are identical. The correct choice is option A.
;
