Answer :
Let's analyze the function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] in detail to identify the correct statements. We'll break down each component to ascertain the function's characteristics such as its domain, range, initial value, and the simplified base.
1. Domain:
- The function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is defined for any real number [tex]\( x \)[/tex]. There are no restrictions on the values [tex]\( x \)[/tex] can take.
- Conclusion: The domain is all real numbers.
2. Range:
- The expression [tex]\( (\sqrt{18})^x \)[/tex] is always positive regardless of the value of [tex]\( x \)[/tex].
- Since we are multiplying by 3, the entire function [tex]\( f(x) \)[/tex] remains positive for all [tex]\( x \)[/tex].
- The smallest value [tex]\( f(x) \)[/tex] can take is when [tex]\( x = 0 \)[/tex], which gives [tex]\( 3 (\sqrt{18})^0 = 3 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows exponentially larger, but never becomes less than 3. Thus, [tex]\( f(x) \)[/tex] is always positive and strictly greater than 0, not necessarily greater than 3.
- Conclusion: The range is [tex]\( y > 0 \)[/tex], not [tex]\( y > 3 \)[/tex].
3. Initial Value:
- The initial value of a function is determined by evaluating it at [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 (\sqrt{18})^0 = 3 \cdot 1 = 3 \][/tex]
- Conclusion: The initial value is [tex]\( 3 \)[/tex].
4. Simplified Base:
- The base of the function, [tex]\( \sqrt{18} \)[/tex], can be simplified.
- [tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
- Therefore, the base becomes simplified as [tex]\( 3 \sqrt{2} \)[/tex], but this alone does not represent the whole base, which is multiplied by an initial factor of 3:
[tex]\[ f(x) = 3 (3 \cdot \sqrt{2})^x = 3 \cdot (9 \sqrt{2}) \][/tex]
- Conclusion: The simplified base in the context of [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is not just [tex]\( 3 \sqrt{2} \)[/tex], but instead the term under the exponent base should be recognized as [tex]\( 3 \sqrt{2} \)[/tex].
Given this breakdown, the correct statements that accurately describe 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].
1. Domain:
- The function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is defined for any real number [tex]\( x \)[/tex]. There are no restrictions on the values [tex]\( x \)[/tex] can take.
- Conclusion: The domain is all real numbers.
2. Range:
- The expression [tex]\( (\sqrt{18})^x \)[/tex] is always positive regardless of the value of [tex]\( x \)[/tex].
- Since we are multiplying by 3, the entire function [tex]\( f(x) \)[/tex] remains positive for all [tex]\( x \)[/tex].
- The smallest value [tex]\( f(x) \)[/tex] can take is when [tex]\( x = 0 \)[/tex], which gives [tex]\( 3 (\sqrt{18})^0 = 3 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] grows exponentially larger, but never becomes less than 3. Thus, [tex]\( f(x) \)[/tex] is always positive and strictly greater than 0, not necessarily greater than 3.
- Conclusion: The range is [tex]\( y > 0 \)[/tex], not [tex]\( y > 3 \)[/tex].
3. Initial Value:
- The initial value of a function is determined by evaluating it at [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 (\sqrt{18})^0 = 3 \cdot 1 = 3 \][/tex]
- Conclusion: The initial value is [tex]\( 3 \)[/tex].
4. Simplified Base:
- The base of the function, [tex]\( \sqrt{18} \)[/tex], can be simplified.
- [tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
- Therefore, the base becomes simplified as [tex]\( 3 \sqrt{2} \)[/tex], but this alone does not represent the whole base, which is multiplied by an initial factor of 3:
[tex]\[ f(x) = 3 (3 \cdot \sqrt{2})^x = 3 \cdot (9 \sqrt{2}) \][/tex]
- Conclusion: The simplified base in the context of [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is not just [tex]\( 3 \sqrt{2} \)[/tex], but instead the term under the exponent base should be recognized as [tex]\( 3 \sqrt{2} \)[/tex].
Given this breakdown, the correct statements that accurately describe 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].
