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The image you uploaded shows the mathematical expression for \( E \):

\[
E := \left\{ \frac{1}{1+x^2} \mid x \in \mathbb{R} \right\}
\]

This defines the set \( E \) as the set of all values of the function \( f(x) = \frac{1}{1+x^2} \), where \( x \in \mathbb{R} \) (i.e., \( x \) is a real number).

### Steps to understand the solution:
1. **Expression**: The function \( f(x) = \frac{1}{1+x^2} \) takes values for all real numbers \( x \).
   
2. **Range of the function**: 
   - The denominator \( 1 + x^2 \) is always positive and greater than or equal to 1, because \( x^2 \geq 0 \).
   - As \( x^2 \to \infty \), \( \frac{1}{1+x^2} \to 0 \), meaning the function approaches 0 but never reaches it.
   - The maximum value occurs when \( x = 0 \), which gives \( f(0) = 1 \).

Thus, the range of the function is \( (0, 1] \). Therefore, the set \( E \) is the interval of values that the function can take, which is:

\[
E = (0, 1]
\]

Would you like more details or have any other questions? Here are some related questions to deepen your understanding:

1. What are the properties of the function \( f(x) = \frac{1}{1+x^2} \)?
2. How do you compute the limit of \( f(x) \) as \( x \to \infty \)?
3. What is the graph of \( f(x) \) and how does it behave?
4. How do you determine the range of other functions involving rational expressions?
5. How would the range of the function change if it were defined over a different domain?

**Tip**: When dealing with rational functions, the denominator's behavior helps you understand the function's range and asymptotic behavior.

Learn how to determine the range of the function f(x) = 1 / (1 + x^2), a rational expression. This problem involves exploring the behavior of the function as x approaches infinity and understanding its maximum and minimum values. Suitable for students in introductory real analysis or advanced calculus.

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