which statements accurately describe the function $f(x)=3(16)^{\frac{3}{4}x}$? select three options.\nthe…

which statements accurately describe the function $f(x)=3(16)^{\frac{3}{4}x}$? select three options.\nthe initial value is 3.\nthe domain is $x > 0$.\nthe range is $y>0$.\nthe simplified base is 12.\nthe simplified base is 8.

which statements accurately describe the function $f(x)=3(16)^{\frac{3}{4}x}$? select three options.\nthe initial value is 3.\nthe domain is $x > 0$.\nthe range is $y>0$.\nthe simplified base is 12.\nthe simplified base is 8.

Answer

Explanation:

Step1: Find the initial value

When (x = 0), (f(0)=3\times(16)^{\frac{3}{4}\times0}=3\times1 = 3).

Step2: Determine the domain

For an exponential - type function (y = a\cdot b^{kx}), the domain of the function (f(x)=3\times(16)^{\frac{3}{4}x}) is all real numbers, i.e., (x\in(-\infty,\infty)), not (x > 0).

Step3: Determine the range

Since (a = 3>0) and (b=(16)^{\frac{3}{4}}>0), the range of the exponential function (y = 3\times(16)^{\frac{3}{4}x}) is (y>0) because the exponential part ((16)^{\frac{3}{4}x}>0) for all real (x) and multiplying by a positive constant (3) still gives a positive result.

Step4: Simplify the base

Calculate ((16)^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^{3}). Since (16^{\frac{1}{4}} = 2) (because (2^{4}=16)), then (\left(16^{\frac{1}{4}}\right)^{3}=2^{3}=8).

Answer:

The initial value is 3. The range is (y > 0). The simplified base is 8.