The given function is:

$f(x) = \frac{1}{x^2 + 1}$

To find the range:

1. Understand the function:

   • The denominator $x^2 + 1$ is always positive because $x^2 \geq 0$ for all real numbers $x$, and adding 1 ensures that the denominator is always at least 1.
   • Therefore, $f(x)$ is always positive and well-defined for all real $x$.

2. Analyze the behavior:

   • As $x \to \infty$ or $x \to -\infty$, the term $x^2$ dominates, and the function approaches $0$ (but never reaches it). Thus, $f(x) \to 0$.
   • At $x = 0$, the function achieves its maximum value: $f(0) = \frac{1}{0^2 + 1} = 1$

3. Conclusion:

   • The function decreases monotonically from $1$ as $x$ moves away from 0, and approaches 0 as $|x| \to \infty$.
   • Hence, the range of the function is: $(0, 1]$

This means the function takes values between 0 (not inclusive) and 1 (inclusive).

Would you like further clarification or details?

Related Questions:

1. How would the range change if the function were $\frac{1}{x^2 - 1}$?
2. What is the domain of the function $f(x) = \frac{1}{x^2 + 1}$?
3. How does the behavior of the function change at extreme values of $x$?
4. Can you sketch the graph of $f(x) = \frac{1}{x^2 + 1}$ to visualize the range?
5. How do you find the range of a rational function in general?

Tip: Always check for any vertical or horizontal asymptotes when analyzing the range of rational functions.