Let's analyze the function [tex]\( f(x) = 3 \cdot 16^{\frac{3}{4} x} \)[/tex] step-by-step to determine which statements about this function are true.
### Step 1: Identify the initial value
- The initial value of the function is the coefficient in front of the exponential expression. In [tex]\( f(x) = 3 \cdot 16^{\frac{3}{4} x} \)[/tex], the coefficient is 3.
- The initial value is 3. This statement is true.
### Step 2: Determine the domain
- The function [tex]\( f(x) = 3 \cdot 16^{\frac{3}{4} x} \)[/tex] is an exponential function. Exponential functions are defined for all real numbers.
- However, there is no restriction in [tex]\( x \)[/tex] that would prohibit any real number from being used. Thus, [tex]\( x \)[/tex] can take any real value.
- The domain is all real numbers, not [tex]\( x > 0 \)[/tex]. Hence, the statement "The domain is [tex]\( x > 0 \)[/tex]" is false.
### Step 3: Determine the range
- For any real number [tex]\( x \)[/tex], [tex]\( 16^{\frac{3}{4} x} \)[/tex] will always be positive because the base 16 is a positive number, and raising it to any power (positive or negative) results in a positive number.
- Multiplying this positive number by 3 will still yield a positive number.
- Therefore, the range of [tex]\( f(x) \)[/tex] is all positive real numbers.
- The range is [tex]\( y > 0 \)[/tex]. This statement is true.
### Step 4: Simplify the base
- To simplify the expression [tex]\( 16^{\frac{3}{4}} \)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8 \][/tex]
- Thus, the simplified base is 8.
- The simplified base is 8. This statement is true.
- The simplified base is not 12. This statement is false.
### Summarizing the accurate statements:
1. The initial value is 3.
2. The range is [tex]\( y > 0 \)[/tex].
3. The simplified base is 8.
So, the three accurate statements describing the function are:
- The initial value is 3.
- The range is [tex]\( y > 0 \)[/tex].
- The simplified base is 8.
