The given expression is:

$f(x)= \frac{1}{2}(27)^{ \frac{2x}{3} } \\ \\ 
f(x)= \frac{1}{2}((27)^{ \frac{2}{3} })^{x} \\ \\ 
f(x)= \frac{1}{2}(9)^{x}$

From the above equation, we can answer all the 4 questions.

a) Initial value occurs when x = 0

So,

$f(0)= \frac{1}{2}(9)^{0}= \frac{1}{2}$

So, the Initial value of the function is 1/2.

b) Simplified Base:
The base of an exponential function is the term which contains the exponent.
From the simplified equation, we can see that the base has been simplified and is equal to 9.

So, the simplified base is 9

c) Domain:
From the equation we can see that there is no such value which makes the function undefined. So all reall numbers are included in the Domain.

So, Domain is Set of All real numbers.
In Interval notation this can be expressed as: (-∞,∞)

d) Range:
Range is the set of all y values that a function can take. Since the graph is not shifted vertically up or down, the range will be the set of all real numbers greater than zero.

So, Range is equal to All real numbers greater than 0.
In Interval notation, this can be expressed as (0,∞)

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