Let’s analyze the function [tex]\( f(x) = 3(16)^{\frac{3}{4} x} \)[/tex] to determine which of the given statements accurately describe it.
### 1. Initial Value
To find the initial value of the function, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \cdot (16)^{\frac{3}{4} \cdot 0} = 3 \cdot 16^0 = 3 \cdot 1 = 3
\][/tex]
So, the initial value of the function is 3. This statement is correct.
### 2. Domain
Next, we consider the domain of the function [tex]\( f(x) \)[/tex]. The base [tex]\( 16 \)[/tex] raised to any real number power is always defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\( \mathbb{R} \)[/tex], not just [tex]\( x > 0 \)[/tex]. This statement is incorrect.
### 3. Range
To find the range of [tex]\( f(x) = 3(16)^{\frac{3}{4} x} \)[/tex], note that [tex]\( (16)^{\frac{3}{4} x} \)[/tex] will always be positive since an exponential function with a positive base (16) raised to any exponent is positive. Therefore, multiplying this by 3 keeps the product positive. Hence, [tex]\( f(x) \)[/tex] is always positive:
[tex]\[
y = 3(16)^{\frac{3}{4} x} > 0
\][/tex]
So, the range of [tex]\( f(x) \)[/tex] is [tex]\( y > 0 \)[/tex]. This statement is correct.
### 4. Simplified Base
To simplify the base of [tex]\( 16^{\frac{3}{4} x} \)[/tex], we find:
[tex]\[
16 = 2^4
\][/tex]
Therefore,
[tex]\[
16^{\frac{3}{4} x} = (2^4)^{\frac{3}{4} x} = 2^{4 \cdot \frac{3}{4} x} = 2^{3x}
\][/tex]
Thus, the base of the simplified expression [tex]\( 2^{3x} \)[/tex] is [tex]\( 2 \)[/tex]. We can now check if it simplifies to either 12 or 8:
Considering the simplified form of 16,
[tex]\[
16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^3 = 8,
\][/tex]
So, the simplified base is 8, not 12.
Hence, the statement “The simplified base is 8” is correct, while the statement “The simplified base is 12” is incorrect.
### Summary
The accurate statements that describe the function [tex]\( f(x) = 3(16)^{\frac{3}{4} x} \)[/tex] are:
1. The initial value is 3.
3. The range is [tex]\( y > 0 \)[/tex].
5. The simplified base is 8.
Thus, the correct options are:
- The initial value is 3.
- The range is [tex]\( y > 0 \)[/tex].
- The simplified base is 8.
