Answer :
Sure! Let's simplify the function [tex]\( f(x) = 7\sqrt[3]{54} \)[/tex] step by step and then answer the questions asked.
1. Simplifying the Base of the Function:
We start with the function [tex]\( f(x) = 7\sqrt[3]{54} \)[/tex].
Notice that:
[tex]\[ 54 = 27 \times 2 \][/tex]
Therefore:
[tex]\[ f(x) = 7 \sqrt[3]{27 \times 2} \][/tex]
We can use the property of cube roots that [tex]\( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \)[/tex]:
[tex]\[ f(x) = 7 \left( \sqrt[3]{27} \times \sqrt[3]{2} \right) \][/tex]
We know from basic mathematics that [tex]\( \sqrt[3]{27} = 3 \)[/tex]:
[tex]\[ f(x) = 7 \times 3 \times \sqrt[3]{2} \][/tex]
Hence:
[tex]\[ f(x) = 21 \sqrt[3]{2} \][/tex]
2. Determining the Initial Value of the Function:
The initial value can be understood as the coefficient or constant factor when there are no variables present. Here, we see that:
[tex]\[ 21 \quad \text{is the initial constant factor} \][/tex]
3. The Simplified Base of the Function:
The term inside the cube root is [tex]\( 2 \)[/tex], and the factor outside is [tex]\( 21 \)[/tex].
Thus, the simplified base is:
[tex]\[ 21 \sqrt[3]{2} \][/tex]
4. Determining the Domain of the Function:
The domain of [tex]\( f(x) \)[/tex] depends on the values of [tex]\( x \)[/tex] that [tex]\( f(x) \)[/tex] can take. In this case, [tex]\( f(x) = 21 \sqrt[3]{2} \)[/tex] does not actually contain [tex]\( x \)[/tex] but let's consider the function and its form. The cube root function is defined for all real numbers.
So, the domain of the function is:
[tex]\[ \text{all real numbers} \][/tex]
5. Determining the Range of the Function:
The range of [tex]\( f(x) \)[/tex] is the set of all possible output values. Since a cube root function can take any real number and produce any real number as an output when multiplied by a non-zero constant [tex]\( 21 \)[/tex], the range of the function is:
[tex]\[ \text{all real numbers} \][/tex]
Thus, the answers to the questions are:
1. The initial value for the function is [tex]\( 21 \)[/tex].
2. The simplified base for the function is [tex]\( 21 \sqrt[3]{2} \)[/tex].
3. The domain of the function is all real numbers.
4. The range of the function is all real numbers.
1. Simplifying the Base of the Function:
We start with the function [tex]\( f(x) = 7\sqrt[3]{54} \)[/tex].
Notice that:
[tex]\[ 54 = 27 \times 2 \][/tex]
Therefore:
[tex]\[ f(x) = 7 \sqrt[3]{27 \times 2} \][/tex]
We can use the property of cube roots that [tex]\( \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \)[/tex]:
[tex]\[ f(x) = 7 \left( \sqrt[3]{27} \times \sqrt[3]{2} \right) \][/tex]
We know from basic mathematics that [tex]\( \sqrt[3]{27} = 3 \)[/tex]:
[tex]\[ f(x) = 7 \times 3 \times \sqrt[3]{2} \][/tex]
Hence:
[tex]\[ f(x) = 21 \sqrt[3]{2} \][/tex]
2. Determining the Initial Value of the Function:
The initial value can be understood as the coefficient or constant factor when there are no variables present. Here, we see that:
[tex]\[ 21 \quad \text{is the initial constant factor} \][/tex]
3. The Simplified Base of the Function:
The term inside the cube root is [tex]\( 2 \)[/tex], and the factor outside is [tex]\( 21 \)[/tex].
Thus, the simplified base is:
[tex]\[ 21 \sqrt[3]{2} \][/tex]
4. Determining the Domain of the Function:
The domain of [tex]\( f(x) \)[/tex] depends on the values of [tex]\( x \)[/tex] that [tex]\( f(x) \)[/tex] can take. In this case, [tex]\( f(x) = 21 \sqrt[3]{2} \)[/tex] does not actually contain [tex]\( x \)[/tex] but let's consider the function and its form. The cube root function is defined for all real numbers.
So, the domain of the function is:
[tex]\[ \text{all real numbers} \][/tex]
5. Determining the Range of the Function:
The range of [tex]\( f(x) \)[/tex] is the set of all possible output values. Since a cube root function can take any real number and produce any real number as an output when multiplied by a non-zero constant [tex]\( 21 \)[/tex], the range of the function is:
[tex]\[ \text{all real numbers} \][/tex]
Thus, the answers to the questions are:
1. The initial value for the function is [tex]\( 21 \)[/tex].
2. The simplified base for the function is [tex]\( 21 \sqrt[3]{2} \)[/tex].
3. The domain of the function is all real numbers.
4. The range of the function is all real numbers.
