Final answer:
To simplify the function f(x) = 1/3(81)^(3x/4), we can rewrite 81 as 3^4. The initial value of the function is 1/3 and the simplified base is 3^(3x). The domain is all real numbers and the range is all positive real numbers.
Step-by-step explanation:
To simplify the function f(x) = 1/3(81)^(3x/4), we can first simplify the base. Since 81 is the same as 3^4, we can rewrite it as 3^4. Now we have f(x) = 1/3(3^4)^(3x/4). We can simplify further by using the property of exponentiation, which states that (a^b)^c is equal to a^(b*c). Applying this property, we have f(x) = 1/3(3^(4*(3x/4))). Simplifying the exponent, we have f(x) = 1/3(3^(3x)).
The initial value of the function is 1/3, which is the coefficient in front of the base.
The simplified base is 3^(3x).
The domain of the function is all real numbers, as there are no restrictions on the input x.
The range of the function is all positive real numbers, as the base is raised to the power of any real number, resulting in a positive output.