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.

Explore the function f(x) = arctan(3^x) with detailed insights into its domain, range, behavior at infinity, and monotonicity. Learn about the derivative of this function and compare it to the arctangent function. Suitable for advanced high school students and beyond.

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