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To analyze the function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex], let's break down the important features.
1. Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is an exponential function, and exponential functions are defined for all real numbers. Thus, the domain is all real numbers.
2. Range: The range of a function is the set of all possible output values (y-values). Given an exponential function of the form [tex]\( a \cdot b^x \)[/tex] with [tex]\( b > 0 \)[/tex], the range depends on the value of [tex]\( b \)[/tex]. For [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex], since the base [tex]\( \sqrt{18} \approx 4.24 \)[/tex] is greater than 1, the function grows rapidly as [tex]\( x \)[/tex] increases and tends towards 0 as [tex]\( x \)[/tex] decreases to negative infinity, but never actually reaches 0 or a negative value. Thus, the range is [tex]\( y > 0 \)[/tex]. Specifically, the statement "The range is [tex]\( y > 3 \)[/tex]" is incorrect; the range is [tex]\( y > 0 \)[/tex].
3. Initial Value: The initial value of the function is found by substituting [tex]\( x = 0 \)[/tex]. For our function:
[tex]\[ f(0) = 3 (\sqrt{18})^0 = 3 \cdot 1 = 3 \][/tex]
Therefore, the initial value is 3.
4. Simplified Base: The base of our exponential function is [tex]\( \sqrt{18} \)[/tex]. We can further simplify this as follows:
[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
Therefore, the simplified base of the function is [tex]\( 3 \sqrt{2} \)[/tex].
Based on the analysis above, the 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].
Summarizing:
- Accurate Statements:
1. The domain is all real numbers.
2. The initial value is 3.
3. The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
- Inaccurate Statements:
- The range is [tex]\( y > 3 \)[/tex]. (The true range is [tex]\( y > 0 \)[/tex])
- The initial value is 9. (The true initial value is 3)
Therefore, the three options selected are:
- The domain is all real numbers.
- The initial value is 3.
- The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
1. Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex] is an exponential function, and exponential functions are defined for all real numbers. Thus, the domain is all real numbers.
2. Range: The range of a function is the set of all possible output values (y-values). Given an exponential function of the form [tex]\( a \cdot b^x \)[/tex] with [tex]\( b > 0 \)[/tex], the range depends on the value of [tex]\( b \)[/tex]. For [tex]\( f(x) = 3 (\sqrt{18})^x \)[/tex], since the base [tex]\( \sqrt{18} \approx 4.24 \)[/tex] is greater than 1, the function grows rapidly as [tex]\( x \)[/tex] increases and tends towards 0 as [tex]\( x \)[/tex] decreases to negative infinity, but never actually reaches 0 or a negative value. Thus, the range is [tex]\( y > 0 \)[/tex]. Specifically, the statement "The range is [tex]\( y > 3 \)[/tex]" is incorrect; the range is [tex]\( y > 0 \)[/tex].
3. Initial Value: The initial value of the function is found by substituting [tex]\( x = 0 \)[/tex]. For our function:
[tex]\[ f(0) = 3 (\sqrt{18})^0 = 3 \cdot 1 = 3 \][/tex]
Therefore, the initial value is 3.
4. Simplified Base: The base of our exponential function is [tex]\( \sqrt{18} \)[/tex]. We can further simplify this as follows:
[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
Therefore, the simplified base of the function is [tex]\( 3 \sqrt{2} \)[/tex].
Based on the analysis above, the 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].
Summarizing:
- Accurate Statements:
1. The domain is all real numbers.
2. The initial value is 3.
3. The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
- Inaccurate Statements:
- The range is [tex]\( y > 3 \)[/tex]. (The true range is [tex]\( y > 0 \)[/tex])
- The initial value is 9. (The true initial value is 3)
Therefore, the three options selected are:
- The domain is all real numbers.
- The initial value is 3.
- The simplified base is [tex]\( 3 \sqrt{2} \)[/tex].
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