Use the function below to find [tex]$f(3)$[/tex]. [tex]$f(x)=\frac{1}{3} \cdot 4^x$[/tex]
A. 4
B. 8
C. [tex]$\frac{16}{3}$[/tex]
D. [tex]$\frac{64}{3}$[/tex]
Answer (2)
Substitute x = 3 into the function f ( x ) = 3 1 ⋅ 4 x .
Calculate 4 3 = 64 .
Multiply the result by 3 1 to get 3 64 .
The final answer is 3 64 .
Explanation
Understanding the problem We are given the function f ( x ) = f r a c 1 3 c d o t 4 x and asked to find the value of f ( 3 ) . This means we need to substitute x = 3 into the function and calculate the result.
Substitution Substitute x = 3 into the function: f ( 3 ) = 3 1 ⋅ 4 3
Calculating the power Calculate 4 3 : 4 3 = 4 ⋅ 4 ⋅ 4 = 16 ⋅ 4 = 64
Final Calculation Multiply the result by f r a c 1 3 : f ( 3 ) = 3 1 ⋅ 64 = 3 64
Final Answer Therefore, f ( 3 ) = f r a c 64 3 .
Examples
Imagine you're baking a cake and the recipe calls for a certain ingredient that doubles in quantity every hour. If you start with 3 1 of a cup of this ingredient, the function f ( x ) = 3 1 ⋅ 2 x tells you how much you'll have after x hours. Similarly, this type of exponential function can model bacterial growth, compound interest, or the decay of radioactive substances. Understanding how to evaluate these functions helps in predicting outcomes in various real-world scenarios.
To find f ( 3 ) using the function f ( x ) = 3 1 ⋅ 4 x , we substitute x = 3 to get f ( 3 ) = 3 64 . The correct answer is D: 3 64 .
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