The function $y = \frac{1}{x^2 + 1}$ is a rational function, because it is the ratio of two polynomials. Specifically, it's the ratio of the constant $1$ over the quadratic expression $x^2 + 1$.

Characteristics of this function:

1. Domain: The function is defined for all real numbers $x$, because the denominator $x^2 + 1$ is always positive and never equals zero (there are no real roots for $x^2 + 1 = 0$).

2. Range: The output (or range) of the function is bounded between $0 < y \leq 1$. The value of $y$ is always positive, but never zero, and the largest value occurs at $x = 0$, where $y = 1$.

3. Asymptotes:

   • Horizontal asymptote: As $x \to \infty$ or $x \to -\infty$, the value of the function approaches $y = 0$, so there is a horizontal asymptote at $y = 0$.
   • Vertical asymptote: There are no vertical asymptotes, since the denominator $x^2 + 1$ never equals zero.

4. Even function: The function is symmetric about the y-axis because $f(x) = f(-x)$. This means it is an even function.

5. Decreasing behavior: The function is strictly decreasing for $x > 0$ and for $x < 0$, meaning it becomes smaller as $|x|$ increases.

Would you like further details on how to derive any of these characteristics or ask any specific questions about the function?

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Here are five related questions:

1. What are the properties of rational functions in general?
2. How do you determine the horizontal asymptote of a rational function?
3. Can you find the first derivative of $y = \frac{1}{x^2 + 1}$ to determine the rate of change?
4. How do we graph rational functions with asymptotes?
5. How do you solve rational inequalities involving similar functions?

Tip: When analyzing rational functions, always check the domain first by ensuring the denominator does not equal zero!