which is the graph of $f(x)=\frac{3}{2}(\frac{1}{3})^x$?

Explanation:

Step1: Analyze the exponential - function form

The general form of an exponential function is (y = ab^{x}), where in (f(x)=\frac{3}{2}(\frac{1}{3})^{x}), (a=\frac{3}{2}) and (b = \frac{1}{3}). Since (0\lt b=\frac{1}{3}\lt1), the function is a decreasing exponential function.

Step2: Find the y - intercept

When (x = 0), (f(0)=\frac{3}{2}(\frac{1}{3})^{0}). According to the rule (a^{0}=1) for (a\neq0), we have (f(0)=\frac{3}{2}\times1=\frac{3}{2}=1.5).

Step3: Check the end - behavior

As (x\rightarrow+\infty), (y = f(x)\rightarrow0) because (0\lt b\lt1). As (x\rightarrow-\infty), (y = f(x)\rightarrow+\infty).

Answer:

The graph that is a decreasing curve and intersects the y - axis at the point ((0,1.5)) and approaches the x - axis as (x) goes to positive infinity.