Answer :
Let's analyze the function [tex]\( f(x) = 3(16)^{\frac{3}{4}} x \)[/tex] to identify which statements accurately describe it.
### Step-by-Step Solution:
1. Initial Value:
- The initial value typically refers to the coefficient of [tex]\( x \)[/tex].
- Here, it is the number immediately before the variable [tex]\( x \)[/tex].
- The initial value of the function is [tex]\( 3 \)[/tex].
2. Domain:
- The domain of a function is the set of all possible input values ([tex]\( x \)[/tex]) for which the function is defined.
- For the function [tex]\( f(x) = 3(16)^{\frac{3}{4}} x \)[/tex], [tex]\( x \)[/tex] must be positive since it is a linear function with no restrictions, other than the implicit assumption of using positive values which is common in many contexts:
- Therefore, the domain is [tex]\( x > 0 \)[/tex].
3. Range:
- The range is the set of all possible output values ([tex]\( y \)[/tex]) produced by the function.
- Since the coefficient of [tex]\( x \)[/tex] (3 multiplied by a positive number) is positive and non-zero, and [tex]\( x \)[/tex] is restricted to positive values:
- The function will always produce positive [tex]\( y \)[/tex] values.
- Hence, the range is [tex]\( y > 0 \)[/tex].
4. Simplified Base:
- We need to simplify [tex]\((16)^{\frac{3}{4}}\)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ (16)^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 * \frac{3}{4}} = 2^3 = 8 \][/tex]
- Therefore, the simplified base is [tex]\( 8 \)[/tex].
### Conclusion:
From the above analysis, we confirm that the function [tex]\( f(x) = 3(16)^{\frac{3}{4}} x \)[/tex] can be accurately described by the following three options:
- The initial value is 3.
- The domain is [tex]\( x > 0 \)[/tex].
- The range is [tex]\( y > 0 \)[/tex].
Hence, the accurate statements are:
1. The initial value is 3.
2. The domain is [tex]\( x > 0 \)[/tex].
3. The range is [tex]\( y > 0 \)[/tex].
4. The simplified base is 8.
These are the accurate descriptions of the given function.
### Step-by-Step Solution:
1. Initial Value:
- The initial value typically refers to the coefficient of [tex]\( x \)[/tex].
- Here, it is the number immediately before the variable [tex]\( x \)[/tex].
- The initial value of the function is [tex]\( 3 \)[/tex].
2. Domain:
- The domain of a function is the set of all possible input values ([tex]\( x \)[/tex]) for which the function is defined.
- For the function [tex]\( f(x) = 3(16)^{\frac{3}{4}} x \)[/tex], [tex]\( x \)[/tex] must be positive since it is a linear function with no restrictions, other than the implicit assumption of using positive values which is common in many contexts:
- Therefore, the domain is [tex]\( x > 0 \)[/tex].
3. Range:
- The range is the set of all possible output values ([tex]\( y \)[/tex]) produced by the function.
- Since the coefficient of [tex]\( x \)[/tex] (3 multiplied by a positive number) is positive and non-zero, and [tex]\( x \)[/tex] is restricted to positive values:
- The function will always produce positive [tex]\( y \)[/tex] values.
- Hence, the range is [tex]\( y > 0 \)[/tex].
4. Simplified Base:
- We need to simplify [tex]\((16)^{\frac{3}{4}}\)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ (16)^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 * \frac{3}{4}} = 2^3 = 8 \][/tex]
- Therefore, the simplified base is [tex]\( 8 \)[/tex].
### Conclusion:
From the above analysis, we confirm that the function [tex]\( f(x) = 3(16)^{\frac{3}{4}} x \)[/tex] can be accurately described by the following three options:
- The initial value is 3.
- The domain is [tex]\( x > 0 \)[/tex].
- The range is [tex]\( y > 0 \)[/tex].
Hence, the accurate statements are:
1. The initial value is 3.
2. The domain is [tex]\( x > 0 \)[/tex].
3. The range is [tex]\( y > 0 \)[/tex].
4. The simplified base is 8.
These are the accurate descriptions of the given function.
