Simplify the function [tex]f(x)=\frac{1}{3}(81)^{\frac{3 x}{4}}[/tex]. Then determine the key aspects of the function.
The initial value is $\square$
The simplified base is $\square$
The domain is $\square$
The range is $\square$
Answer (2)
Simplify the function: Rewrite 81 as 3 4 and use exponent rules to get f ( x ) = 3 1 ( 27 ) x .
Find the initial value: Evaluate f ( 0 ) = 3 1 ( 27 ) 0 = 3 1 .
Identify the base: The simplified base is 27 .
Determine the domain and range: The domain is ( − ∞ , ∞ ) and the range is ( 0 , ∞ ) .
The initial value is 3 1 , the simplified base is 27 , the domain is ( − ∞ , ∞ ) , and the range is ( 0 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 3 1 ( 81 ) 4 3 x and we need to simplify it and find its key aspects: the initial value, the simplified base, the domain, and the range.
Simplifying the Function First, let's simplify the function. We know that 81 = 3 4 , so we can rewrite the function as: f ( x ) = 3 1 ( 3 4 ) 4 3 x Using the property of exponents, ( a b ) c = a b ⋅ c , we have: f ( x ) = 3 1 ⋅ 3 4 ⋅ 4 3 x = 3 1 ⋅ 3 3 x We can further simplify this by rewriting 3 3 x as ( 3 3 ) x = 2 7 x . Thus, f ( x ) = 3 1 ( 27 ) x
Finding the Initial Value Now, let's find the initial value. The initial value is the value of the function when x = 0 . So, f ( 0 ) = 3 1 ( 27 ) 0 = 3 1 ( 1 ) = 3 1 Thus, the initial value is 3 1 .
Identifying the Simplified Base The simplified base is the base of the exponential function, which is 27.
Determining the Domain The domain of an exponential function is all real numbers, since we can plug in any real number for x . Therefore, the domain is ( − ∞ , ∞ ) .
Determining the Range The range of the function is the set of all possible output values. Since the base 27 is positive and the coefficient 3 1 is positive, the range is all positive real numbers. Therefore, the range is ( 0 , ∞ ) .
Final Answer In summary:
The simplified function is f ( x ) = 3 1 ( 27 ) x .
The initial value is 3 1 .
The simplified base is 27.
The domain is ( − ∞ , ∞ ) .
The range is ( 0 , ∞ ) .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that earns compound interest, the amount of money you have in the account can be modeled by an exponential function. The initial value represents the initial investment, the base represents the growth factor, and the exponent represents the time period. Understanding exponential functions can help you make informed decisions about your finances and other aspects of your life.
The simplified function is f ( x ) = 3 1 ( 27 ) x with an initial value of 3 1 , a simplified base of 27 , a domain of ( − ∞ , ∞ ) , and a range of ( 0 , ∞ ) .
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