The problem involves graphing the exponential function:

$g(x) = \frac{1}{3} (3)^x$

Step-by-Step Solution:

1. Simplify the Function: The function simplifies to: $g(x) = \frac{1}{3} \cdot 3^x = 3^{x - 1}$ This is an exponential function with a base of $3$, shifted by $x - 1$.

2. Choose $x$-values and Calculate $g(x)$: To graph the function, compute $g(x)$ for five points:

   • $x = -2$: $g(-2) = \frac{1}{3} \cdot 3^{-2} = \frac{1}{3} \cdot \frac{1}{9} = \frac{1}{27}$
   • $x = -1$: $g(-1) = \frac{1}{3} \cdot 3^{-1} = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$
   • $x = 0$: $g(0) = \frac{1}{3} \cdot 3^0 = \frac{1}{3} \cdot 1 = \frac{1}{3}$
   • $x = 1$: $g(1) = \frac{1}{3} \cdot 3^1 = \frac{1}{3} \cdot 3 = 1$
   • $x = 2$: $g(2) = \frac{1}{3} \cdot 3^2 = \frac{1}{3} \cdot 9 = 3$

3. Plot the Points: The points to plot on the graph are: $(-2, \frac{1}{27}), \, (-1, \frac{1}{9}), \, (0, \frac{1}{3}), \, (1, 1), \, (2, 3)$

4. Graph Behavior:

   • For $x \to -\infty$, $g(x) \to 0$ (the graph approaches the horizontal asymptote $y = 0$).
   • For $x \to \infty$, $g(x) \to \infty$ (the graph increases exponentially).

Would you like a plot of the graph or additional clarification?

Related Questions:

1. How does the base of an exponential function affect its growth or decay?
2. What is the effect of the coefficient $\frac{1}{3}$ in the function $g(x) = \frac{1}{3} (3)^x$?
3. How would the graph change if the base were $\frac{1}{3}$ instead of $3$?
4. What is the domain and range of the function $g(x) = \frac{1}{3}(3)^x$?
5. How can transformations (shifts, stretches) affect exponential graphs?

Tip:

When graphing exponential functions, calculate and plot a few key points around $x = 0$ to capture the graph's behavior. These functions often include asymptotes and rapid growth/decay depending on the base.