You have provided two functions to analyze:

27. $f(x) = \frac{1}{x^2 + 1}$ on the interval $[-1, 1]$
28. $f(x) = \frac{1}{x}$ on the interval $[1, e]$

Let me know if you would like me to solve a specific problem related to these functions, such as integration, finding derivatives, analyzing their behavior, or any other operation!

For now, here are five related questions:

1. What is the derivative of $f(x) = \frac{1}{x^2 + 1}$ on $[-1, 1]$?
2. Can you calculate the integral of $f(x) = \frac{1}{x^2 + 1}$ on $[-1, 1]$?
3. What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 1 from the right on $[1, e]$?
4. How does the function $f(x) = \frac{1}{x^2 + 1}$ behave in terms of concavity and critical points?
5. Can we find the area under the curve for $f(x) = \frac{1}{x}$ on $[1, e]$?

Tip: For rational functions, it's useful to identify key points such as limits, vertical asymptotes, and intervals of increasing/decreasing behavior to understand their overall structure.