Which statements accurately describe the function $f(x)=3(\sqrt{18})^x$ ? Select three options.
The domain is all real numbers.
The range is $y>3$.
The initial value is 3.
The initial value is 9.
The simplified base is $3 \sqrt{2}$.
Answer (2)
The domain of the function f ( x ) = 3 ( 18 ) x is all real numbers.
The initial value of the function is f ( 0 ) = 3 .
The simplified base of the function is 18 = 3 2 .
Therefore, the three accurate statements are: The domain is all real numbers, the initial value is 3, and the simplified base is 3 2 . T h e d o main i s a ll re a l n u mb ers , t h e ini t ia l v a l u e i s 3 , an d t h e s im pl i f i e d ba se i s 3 2
Explanation
Understanding the Function 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. Since f ( x ) is an exponential function, its domain is all real numbers.
Determining the Range The range of an exponential function of the form a x is 0"> y > 0 if 0"> a > 0 and 0"> b > 0 . In our case, f ( x ) = 3 ( 18 ) x . Since 0"> 3 > 0 and 0"> 18 > 0 , the range is 0"> y > 0 . However, since the function is 3 ( 18 ) x , the range is actually 0"> y > 0 . Since the coefficient in front of the exponential is 3, the range is 0"> y > 0 .
Finding the Initial Value The initial value of a function is the value of the function when x = 0 . So, we need to find f ( 0 ) .
f ( 0 ) = 3 ( 18 ) 0 = 3 ( 1 ) = 3 Thus, the initial value is 3.
Simplifying the Base To simplify the base, we have 18 = 9 × 2 = 9 × 2 = 3 2 . So, the simplified base is 3 2 .
Checking the Statements Now, let's check the given statements:
The domain is all real numbers. This is true.
The range is 3"> y > 3 . This is false, the range is 0"> y > 0 .
The initial value is 3. This is true.
The initial value is 9. This is false.
The simplified base is 3 2 . This is true.
Final Answer Therefore, the three accurate statements 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 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 after a certain number of years can be modeled by an exponential function. Understanding the properties of exponential functions, such as the domain, range, and initial value, can help you make informed decisions about your investments.
The accurate statements regarding the function f ( x ) = 3 ( s q r t 18 ) x are: the domain is all real numbers, the initial value is 3, and the simplified base is 3 s q r t 2 .
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