The task asks to select the decreasing exponential function among the given ones and plot its graph.

The provided functions are:

1. $f(x) = x^2$
2. $g(x) = \left( \frac{1}{3} \right)^x$
3. $h(x) = 3^x$

Solution:

• $f(x) = x^2$ is a quadratic function, not exponential, and it increases for $x > 0$.
• $g(x) = \left( \frac{1}{3} \right)^x$ is an exponential function with a base $\frac{1}{3}$ (where $0 < \frac{1}{3} < 1$). Exponential functions with a base less than $1$ are decreasing.
• $h(x) = 3^x$ is an exponential function with a base greater than $1$, so it is increasing.

Conclusion:

The decreasing exponential function is: $g(x) = \left( \frac{1}{3} \right)^x$

Would you like me to plot the graph or provide further explanations? Let me know!

Related Questions:

1. What are the general properties of exponential functions?
2. How do we determine if a function is increasing or decreasing?
3. What is the difference between quadratic and exponential growth?
4. How can you rewrite exponential functions in logarithmic form?
5. How does the base of an exponential function affect its behavior?

Tip:

Exponential functions with a base $a$ behave as follows:

• If $a > 1$, the function increases.
• If $0 < a < 1$, the function decreases.