The task requires graphing the exponential function:

$g(x) = \left(\frac{3}{5}\right)^x$

To do this, let us calculate values of $g(x)$ for a set of $x$-values to plot five points on the graph. I'll use typical points near zero to illustrate the behavior of the exponential function.

Calculations:

1. When $x = -2$:
   $g(-2) = \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9} \approx 2.78$

2. When $x = -1$:
   $g(-1) = \left(\frac{3}{5}\right)^{-1} = \frac{5}{3} \approx 1.67$

3. When $x = 0$:
   $g(0) = \left(\frac{3}{5}\right)^0 = 1$

4. When $x = 1$:
   $g(1) = \left(\frac{3}{5}\right)^1 = \frac{3}{5} \approx 0.6$

5. When $x = 2$:
   $g(2) = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \approx 0.36$

Plot Points:

The calculated points are: $(-2, 2.78), \, (-1, 1.67), \, (0, 1), \, (1, 0.6), \, (2, 0.36)$

Graph Behavior:

• The function $g(x)$ decreases as $x$ increases because the base $\frac{3}{5}$ is less than 1.
• For negative $x$, the function grows since it effectively flips the base to $\frac{5}{3}$ due to the negative exponent.

Would you like me to create a visualization of the graph or elaborate on the steps?

5 Related Questions:

1. What does it mean for an exponential function when the base is less than 1?
2. How do negative exponents affect the graph of a function?
3. How can transformations like vertical shifts impact the graph of $g(x)$?
4. Why is $g(0)$ always 1 for any exponential function $a^x$?
5. How do we interpret the asymptote of $g(x) = \left(\frac{3}{5}\right)^x$?

Tip:

When plotting exponential functions, focus on how the base affects growth or decay and remember symmetry for negative exponents.