https://file.aiquickdraw.com/mathbot/file/1726759137848j0x192wl.jpg

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.

Learn how to find the range of the function f(x) = 1/(x^2 + 1). This problem involves analyzing the behavior of a rational function and understanding its range for all real numbers. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation