Determine the domain and range of the function [tex]f(x)=2 \sqrt[3]{108}^{2 x}[/tex].
A. [tex] \{x \mid[/tex] all real numbers); [tex] \{y \mid y\ \textgreater \ 0\}[/tex]
B. [tex] \{x-[/tex] all real numbers): [tex] \{y \mid y \geq 0\}[/tex]
C. (x|x>0): (y| all real numbers)
D. [tex] \{x \mid x \geq 0\} ;\{y \mid[/tex] all real numbers [tex]\}[/tex]
Answer (2)
The domain of the function f ( x ) = 2 3 108 2 x is all real numbers.
Simplify the function to f ( x ) = 2 ( 18 3 2 ) x .
The range of the function is all positive real numbers since the exponential part is always positive.
Therefore, the domain is { x ∣ all real numbers} and the range is { 0"> y ∣ y > 0 }. 0)}}"> ( x ∣ all real numbers ) ; ( y ∣ y > 0 )
Explanation
Understanding the Problem We are asked to find the domain and range of the function f ( x ) = 2 3 108 2 x . Let's analyze the function to determine these properties.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a cube root and an exponential function. Cube roots are defined for all real numbers, and exponential functions are also defined for all real numbers. Therefore, there are no restrictions on the values of x. The domain is all real numbers.
Simplifying the Function The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range, let's first simplify the expression. We have 3 108 = 3 27 ⋅ 4 = 3 3 4 . So, f ( x ) = 2 ( 3 3 4 ) 2 x = 2 ( 3 2 ( 3 4 ) 2 ) x = 2 ( 9 3 16 ) x = 2 ( 9 ⋅ 2 3 2 ) x = 2 ( 18 3 2 ) x .
Analyzing the Range Since 18 3 2 is a positive number, the exponential function ( 18 3 2 ) x will always be positive for any real number x. Therefore, 2 ( 18 3 2 ) x will also always be positive.
Determining the Range As x approaches − ∞ , the value of ( 18 3 2 ) x approaches 0, but it never actually reaches 0. Therefore, f ( x ) approaches 0 but never reaches 0. As x approaches ∞ , the value of ( 18 3 2 ) x approaches ∞ , so f ( x ) also approaches ∞ . Thus, the range of the function is all positive real numbers.
Final Answer Therefore, the domain of the function is all real numbers, and the range is all positive real numbers. In set notation, the domain is { x ∣ all real numbers} and the range is { 0"> y ∣ y > 0 }.
Examples
Exponential functions, like the one in this problem, are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the domain and range of such functions helps us determine the possible values for the input (e.g., time) and the output (e.g., population size). For instance, if we're modeling the growth of a bacteria colony, the domain would represent the time interval over which the model is valid, and the range would represent the possible sizes of the colony. Knowing these limits allows us to make accurate predictions and informed decisions based on the model.
The domain of the function f ( x ) = 2 3 108 2 x is all real numbers, while the range is all positive real numbers. Therefore, the correct answer is option A: 0 \}"> { x ∣ all real numbers } ; { y ∣ y > 0 } .
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