To determine which statements accurately describe the function [tex]\( f(x) = 3(\sqrt{18})^x \)[/tex], we need to analyze the properties of this function step by step.
1. Domain:
The function [tex]\( f(x) = 3(\sqrt{18})^x \)[/tex] is an exponential function. The base, [tex]\(\sqrt{18}\)[/tex], is a positive number. Exponential functions of the form [tex]\( a \cdot b^x \)[/tex] (where [tex]\( b > 0 \)[/tex] and [tex]\( b \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex].
Statement: The domain is all real numbers.
2. Range:
For [tex]\( f(x) = 3(\sqrt{18})^x \)[/tex], since [tex]\(\sqrt{18}\)[/tex] is a positive number, [tex]\((\sqrt{18})^x\)[/tex] will always be positive for all real [tex]\( x \)[/tex]. Additionally, multiplying by 3 scales this positive value but does not change its positivity. Therefore, the function [tex]\( f(x) \)[/tex] will always be positive for any real number [tex]\( x \)[/tex]. Hence, the range is [tex]\( y > 0 \)[/tex], not [tex]\( y > 3 \)[/tex].
Statement: The range is [tex]\( y > 3 \)[/tex] (incorrect).
3. Initial Value:
The initial value of a function refers to the value when [tex]\( x = 0 \)[/tex]. Evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3(\sqrt{18})^0 = 3 \cdot 1 = 3
\][/tex]
Therefore, the initial value of the function is 3.
Statement: The initial value is 3.
4. Initial Value (Alternative):
We already established that when [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 3 \)[/tex]. Therefore, the statement that the initial value is 9 is incorrect.
Statement: The initial value is 9 (incorrect).
5. Simplified Base:
The base of the exponent is [tex]\(\sqrt{18}\)[/tex]. We can simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\][/tex]
Thus, the simplified base is [tex]\( 3\sqrt{2} \)[/tex].
Statement: The simplified base is [tex]\( 3\sqrt{2} \)[/tex].
Putting it all together, the correct statements accurately describing the function [tex]\( f(x) = 3(\sqrt{18})^x \)[/tex] are:
- The domain is all real numbers.
- The initial value is 3.
- The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
Hence, the three correct options are:
1. The domain is all real numbers.
2. The initial value is 3.
3. The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
