Determine which function produces the same graph as [tex]f(x)=\left(8^{\frac{2}{3} x}\right)\left(16^{\frac{1}{2} x}\right)[/tex].
A. [tex]f(x)=4^x[/tex]
B. [tex]f(x)=4^{2 x}[/tex]
C. [tex]f(x)=8^{3 x}[/tex]
D. [tex]f(x)=16^{2 x}[/tex]
Answer (2)
Express 8 and 16 as powers of 2: 8 = 2 3 and 16 = 2 4 .
Rewrite f ( x ) using the base 2: f ( x ) = ( 2 3 ⋅ 3 2 x ) ( 2 4 ⋅ 2 1 x ) = ( 2 2 x ) ( 2 2 x ) .
Simplify the expression using exponent rules: f ( x ) = 2 2 x ⋅ 2 2 x = 2 4 x = ( 2 2 ) 2 x = 4 2 x .
The equivalent function is f ( x ) = 4 2 x .
Explanation
Problem Analysis We are given the function f ( x ) = ( 8 3 2 x ) ( 1 6 2 1 x ) and we want to find an equivalent function from the given options: f ( x ) = 4 x , f ( x ) = 4 2 x , f ( x ) = 8 3 x , f ( x ) = 1 6 2 x .
Expressing in Base 2 First, we express 8 and 16 as powers of 2: 8 = 2 3 and 16 = 2 4 . Substituting these into the expression for f ( x ) , we get f ( x ) = ( ( 2 3 ) 3 2 x ) ( ( 2 4 ) 2 1 x ) Using the power of a power rule, ( a m ) n = a mn , we have f ( x ) = ( 2 3 ⋅ 3 2 x ) ( 2 4 ⋅ 2 1 x ) = ( 2 2 x ) ( 2 2 x )
Simplifying the Expression Now, we use the product of powers rule, a m ⋅ a n = a m + n , to simplify the expression: f ( x ) = 2 2 x ⋅ 2 2 x = 2 2 x + 2 x = 2 4 x We want to express this as a power of 4. Since 4 = 2 2 , we can write 2 = 4 2 1 . Substituting this into our expression, we get f ( x ) = 2 4 x = ( 2 2 ) 2 x = 4 2 x Thus, f ( x ) = 4 2 x .
Comparing with Options Comparing our simplified function f ( x ) = 4 2 x with the given options, we see that it matches the second option, f ( x ) = 4 2 x .
Final Answer Therefore, the function that produces the same graph as f ( x ) = ( 8 3 2 x ) ( 1 6 2 1 x ) is f ( x ) = 4 2 x .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. In finance, understanding exponential growth helps in calculating investment returns over time. For instance, if an investment grows at a rate of r per year, the value of the investment after t years can be modeled by an exponential function. Simplifying and comparing exponential functions, as we did in this problem, allows us to analyze and compare different investment options to determine which one yields the highest return.
The function that produces the same graph as f ( x ) = ( 8 3 2 x ) ( 1 6 2 1 x ) is f ( x ) = 4 2 x , which corresponds to option B.
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