Which statements accurately describe the function $f(x)=3(\sqrt{18})^x$ ? Select three options.
A. The domain is all real numbers.
B. The range is $y>3$.
C. The initial value is 3.
D. The initial value is 9.
E. The simplified base is $3 \sqrt{2}$.
Answer (1)
The domain of the function f ( x ) = 3 ( 18 ) x is all real numbers.
The range of the function is 0"> y > 0 .
The initial value of the function is f ( 0 ) = 3 .
The simplified base of the function is 18 = 3 2 .
Therefore, the correct statements are: The domain is all real numbers, The initial value is 3, and The simplified base is 3 2 .
The final answer is: The domain is all real numbers, The initial value is 3, The simplified base is 3 2
Explanation
Understanding the Problem We are given the function f ( x ) = 3 ( 18 ) x and asked to determine which of the given statements are true. The statements concern the domain, range, initial value, and simplified base of the function.
Determining the Domain The domain of an exponential function is all real numbers. Therefore, the statement 'The domain is all real numbers' is true.
Determining the Range The range of an exponential function of the form a ⋅ b x where 0"> b > 0 is 0"> y > 0 if 0"> a > 0 and y < 0 if a < 0 . In our case, f ( x ) = 3 ( 18 ) x , so 0"> a = 3 > 0 . Thus, the range is 0"> y > 0 . The statement 'The range is 3"> y > 3 ' is false.
Determining the Initial Value The initial value of the function is the value of f ( x ) when x = 0 . So, we need to calculate f ( 0 ) = 3 ( 18 ) 0 = 3 ⋅ 1 = 3 . Therefore, the statement 'The initial value is 3' is true, and the statement 'The initial value is 9' is false.
Simplifying the Base We can simplify the base 18 as follows: 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 . Therefore, the statement 'The simplified base is 3 2 ' is true.
Final Answer The statements that accurately describe the function f ( x ) = 3 ( 18 ) x are:
The domain is all real numbers.
The initial value is 3.
The simplified base is 3 2 .
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 an account that earns compound interest, the amount of money in the account will grow exponentially over time. The function f ( x ) = 3 ( 18 ) x could represent the amount of money in an account after x years, where the initial investment is 3 an d t h e ann u a l g ro wt h r a t e i sre l a t e d t o \sqrt{18}$. Understanding the properties of exponential functions, such as domain, range, and initial value, is crucial for making informed decisions about investments and other real-world applications.
