https://r2.erweima.ai/mathbot/image/1723394334501tjq2t5w9.jpg

To provide the set builder notation for the domain and range of the function shown in the graph, let's first clarify what set builder notation is:

- **Domain**: The domain in set builder notation specifies the set of all possible \( x \) values for which the function is defined.
- **Range**: The range in set builder notation specifies the set of all possible \( y \) values that the function can take.

### Domain in Set Builder Notation
From the graph, it appears that the function is defined for all real numbers. In set builder notation, the domain would be expressed as:

\[
\text{Domain} = \{ x \mid x \in \mathbb{R} \}
\]

This reads as "the set of all \( x \) such that \( x \) is a real number."

### Range in Set Builder Notation
Similarly, the range of the function appears to include all real numbers based on the graph. In set builder notation, the range is:

\[
\text{Range} = \{ y \mid y \in \mathbb{R} \}
\]

This reads as "the set of all \( y \) such that \( y \) is a real number."

**Summary:**
- **Domain** in set builder notation: \( \{ x \mid x \in \mathbb{R} \} \)
- **Range** in set builder notation: \( \{ y \mid y \in \mathbb{R} \} \)

Do you have any other questions or would you like to dive into any other aspect of this problem?

Here are some additional questions to consider:
1. How do you write inequalities in set builder notation?
2. What does the graph tell us about the function's continuity?
3. How would you describe this function's behavior as \( x \) approaches infinity?
4. How can we determine the symmetry of the function using set builder notation?
5. Could this graph represent a trigonometric function, and if so, which one?
6. What transformations could change the domain or range of the function?
7. How would the set builder notation change if the function had restrictions on its domain?
8. How can you use the graph to determine the maximum and minimum values of the function?

**Tip:** Set builder notation is a powerful way to describe sets, especially when dealing with domains and ranges of functions. It helps in clearly defining the conditions under which elements belong to a set.

Explore set builder notation for defining the domain and range of functions with a detailed explanation. Learn how to express domains and ranges using set notation, focusing on real numbers. Suitable for Grades 10-12 mathematics students.

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