Simplify the function [tex]f(x)=\frac{1}{2}(27)^{\frac{2 x}{3}}[/tex]. Then determine the key aspects of the function.
The initial value is
The simplified base is
The domain is $\square$
The range is $\square$
Answer (1)
Simplify the function: Rewrite 27 as 3 3 and use exponent rules to get f ( x ) = 2 1 ( 9 x ) .
Find the initial value: Evaluate f ( 0 ) = 2 1 ( 9 0 ) = 2 1 .
Identify the domain and range: The domain is all real numbers ( − ∞ , ∞ ) , and the range is ( 0 , ∞ ) .
State the key aspects: Initial value is 2 1 , simplified base is 9, domain is ( − ∞ , ∞ ) , and range is ( 0 , ∞ ) . The final answers are: Initial value is 2 1 , simplified base is 9 , domain is ( − ∞ , ∞ ) , and range is ( 0 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 1 ( 27 ) 3 2 x and we want to simplify it and determine its key aspects: initial value, simplified base, domain, and range.
Rewriting the Function First, let's simplify the function. We can rewrite 27 as 3 3 , so the function becomes f ( x ) = 2 1 ( 3 3 ) 3 2 x .
Simplifying the Exponent Using the property of exponents ( a b ) c = a b c , we can simplify the exponent: 3 ⋅ 3 2 x = 2 x . So the function becomes f ( x ) = 2 1 ( 3 2 x ) .
Further Simplification We can rewrite 3 2 x as ( 3 2 ) x = 9 x , so the function simplifies to f ( x ) = 2 1 ( 9 x ) .
Finding the Initial Value Now, let's find the initial value. The initial value is the value of the function when x = 0 : f ( 0 ) = 2 1 ( 9 0 ) = 2 1 ( 1 ) = 2 1 .
Identifying the Simplified Base The simplified base of the exponential function is 9.
Determining the Domain The domain of an exponential function is all real numbers, so the domain of f ( x ) is ( − ∞ , ∞ ) .
Determining the Range The range of 9 x is ( 0 , ∞ ) . Since f ( x ) = 2 1 ( 9 x ) , the range of f ( x ) is ( 0 , ∞ ) .
Final Answer In summary:
The initial value is 2 1 .
The simplified base is 9.
The domain is ( − ∞ , ∞ ) .
The range is ( 0 , ∞ ) .
Examples
Exponential functions are used to model population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns interest compounded continuously, the amount of money you have after a certain time can be modeled by an exponential function. Understanding the base and initial value of the function can help you predict how your investment will grow over time. Similarly, in biology, exponential functions can describe the growth of a bacterial population under ideal conditions. The initial value represents the starting population size, and the base reflects the growth rate.
