which is the graph of the sequence defined by the function $f(x + 1)=\frac{2}{3}f(x)$ if the initial value…

Explanation:

Step1: Identify the initial - value

The initial value of the sequence is given as (f(0)=108) since when (x = 0), we start the sequence.

Step2: Find the first - term of the sequence

Use the recurrence relation (f(x + 1)=\frac{2}{3}f(x)). When (x = 0), (f(1)=\frac{2}{3}f(0)). Substitute (f(0)=108) into the formula: (f(1)=\frac{2}{3}\times108 = 72).

Step3: Find the second - term of the sequence

When (x = 1), (f(2)=\frac{2}{3}f(1)). Substitute (f(1)=72) into the formula: (f(2)=\frac{2}{3}\times72 = 48).

Step4: Find the third - term of the sequence

When (x = 2), (f(3)=\frac{2}{3}f(2)). Substitute (f(2)=48) into the formula: (f(3)=\frac{2}{3}\times48 = 32).

The points ((0,108)), ((1,72)), ((2,48)), ((3,32)) should be on the graph of the sequence. So the correct graph is the one with these points.

Answer:

The graph with points ((0,108)), ((1,72)), ((2,48)), ((3,32))