Which statements accurately describe the function [tex]f(x)=3(16)^{\frac{3}{4} x}[/tex]? Select three options.
A. The initial value is 3.
B. The domain is [tex]x\ \textgreater \ 0[/tex].
C. The range is [tex]y\ \textgreater \ 0[/tex].
D. The simplified base is 12.
E. The simplified base is 8.
Answer (2)
The initial value is found by evaluating f ( 0 ) , which equals 3.
The domain of the function is all real numbers, but the range is 0"> y > 0 because the exponential term is always positive.
The base is simplified as 1 6 4 3 = 8 .
Therefore, the correct statements are: initial value is 3, range is 0"> y > 0 , and the simplified base is 8. 0, \ and \ the \ simplified \ base \ is \ 8.}"> T h e ini t ia l v a l u e i s 3 , t h e r an g e i s y > 0 , an d t h e s im pl i f i e d ba se i s 8.
Explanation
Analyzing the Problem We are given the function f ( x ) = 3 ( 16 ) 4 3 x and asked to select three accurate descriptions of it from the given options. The options are:
The initial value is 3.
The domain is 0"> x > 0 .
The range is 0"> y > 0 .
The simplified base is 12.
The simplified base is 8.
Finding the Initial Value The initial value of a function is the value of the function when x = 0 . Let's calculate f ( 0 ) :
f ( 0 ) = 3 ( 16 ) 4 3 ( 0 ) = 3 ( 16 ) 0 = 3 ( 1 ) = 3 So, the initial value is indeed 3.
Determining the Domain The domain of an exponential function is all real numbers, unless there are restrictions. In this case, there are no restrictions on x , so the domain is all real numbers, not just 0"> x > 0 .
Determining the Range Since the base of the exponential function, 1 6 4 3 x , is always positive, and we are multiplying it by 3, the range of the function will be all positive real numbers. Thus, the range is 0"> y > 0 .
Simplifying the Base To simplify the base, we can rewrite the function as: f ( x ) = 3 ( 16 ) 4 3 x = 3 (( 16 ) 4 3 ) x Now, let's simplify 1 6 4 3 :
1 6 4 3 = ( 2 4 ) 4 3 = 2 4 × 4 3 = 2 3 = 8 So, the simplified base is 8.
Conclusion Based on our analysis, the accurate statements are:
The initial value is 3.
The range is 0"> y > 0 .
The simplified base is 8.
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 3000 inana cco u n tt ha t e a r n s 8 V(t) = 3000(1 + 0.08/4)^{4t} = 3000(1.02)^{4t}$. Understanding the initial value, domain, range, and base of this function can help you predict the future value of your investment.
The accurate statements regarding the function f ( x ) = 3 ( 16 ) 4 3 x are that the initial value is 3, the range is 0"> y > 0 , and the simplified base is 8. The domain includes all real numbers, not just positive values. The selected options are A, C, and E.
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