Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.