The function given is $f(x) = \tan^{-1}(3^x)$. This is an inverse tangent function with the argument $3^x$.

To understand the behavior of this function, consider the following:

1. Domain: The function $\tan^{-1}(y)$ is defined for all real numbers $y$. Therefore, the domain of $f(x) = \tan^{-1}(3^x)$ is all real numbers $x$.

2. Range: The range of $\tan^{-1}(y)$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Thus, the range of $f(x) = \tan^{-1}(3^x)$ is also $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

3. Behavior at Infinity:

   • As $x \to -\infty$, $3^x \to 0$, so $\tan^{-1}(3^x) \to \tan^{-1}(0) = 0$.
   • As $x \to \infty$, $3^x \to \infty$, so $\tan^{-1}(3^x) \to \tan^{-1}(\infty) = \frac{\pi}{2}$.

4. Monotonicity: Since the derivative of $\tan^{-1}(y)$ is $\frac{1}{1 + y^2}$, and $3^x$ is strictly increasing, the function $f(x) = \tan^{-1}(3^x)$ is strictly increasing.

Would you like more details on any specific aspect of this function? Here are five questions you might consider:

1. What are the key properties of the function $\tan^{-1}(3^x)$?
2. How does the function $f(x) = \tan^{-1}(3^x)$ behave as $x$ approaches negative and positive infinity?
3. What is the derivative of the function $f(x) = \tan^{-1}(3^x)$?
4. Can you graph $f(x) = \tan^{-1}(3^x)$?
5. How does the function $f(x) = \tan^{-1}(3^x)$ compare to $y = \tan^{-1}(x)$?

Tip: For a function involving an exponential and an inverse trigonometric function, always consider the behavior at the extremes of the domain to understand its range and limits.