Finding the Domain and Range of a Graph*Determine the domain and range for the graph below. Write your answer in interval notation and in set builder form using a compound inequality.*

***Note:**To type in the ≤ sign, type **<=** or use the Math Quill by clicking the yellow arrow.*                                      Range written in interval notation:__________      Range written in set builder form
Use a compound inequality {y|         }                           the coordinates start on the left at (-5,-4) and the end point is not filled in it then reaches to the right and stops at (5,-5) but the end point IS filled in 

To determine the domain and range of the graph based on the given information, we can break the problem down as follows:

### Domain:
The domain represents the set of all possible \(x\)-values for which the function is defined. Based on the description:
- The graph starts at the point \((-5, -4)\) and continues to the right.
- The graph stops at the point \((5, -5)\), where the endpoint is **filled in**, indicating that \(x = 5\) is included in the domain.
- The point \((-5, -4)\) has an **open circle**, meaning that \(x = -5\) is not included in the domain.

So, the domain in interval notation is:
\[
\text{Domain: } (-5, 5]
\]
In set-builder form, the domain is:
\[
\{x \mid -5 < x \leq 5\}
\]

### Range:
The range represents the set of all possible \(y\)-values for which the function is defined. Based on the description:
- The graph has a lowest \(y\)-value of \(-5\) (since it stops at the point \((5, -5)\), and the endpoint is filled in, meaning \(-5\) is included in the range).
- The highest \(y\)-value is \(-4\) (corresponding to the point \((-5, -4)\)), but this point is open, meaning \(-4\) is **not** included in the range.

So, the range in interval notation is:
\[
\text{Range: } (-5, -4]
\]
In set-builder form, the range is:
\[
\{y \mid -5 < y \leq -4\}
\]

### Final Answer:
- **Domain (interval notation):** \((-5, 5]\)
- **Domain (set-builder notation):** \(\{x \mid -5 < x \leq 5\}\)
- **Range (interval notation):** \((-5, -4]\)
- **Range (set-builder notation):** \(\{y \mid -5 < y \leq -4\}\)

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Would you like a detailed explanation of how interval and set-builder notation work?

### Related Questions:
1. How do you find the domain of a function that includes square roots or fractions?
2. What changes if the graph has no open or closed endpoints?
3. How does the vertical line test relate to domain and range?
4. How can you determine domain and range from an equation without graphing?
5. What does it mean when a function has an "infinite" domain or range?

**Tip:** When using set-builder notation, pay close attention to whether a point is included or not; open and closed circles on a graph will help guide you!

Finding the Domain and Range of a Graph. Determine the domain and range for the graph below. Write your answer in interval notation and in set builder form using a compound inequality.

Learn how to determine the domain and range of a graph using interval notation and set-builder notation. This tutorial covers interpreting graph endpoints, understanding open and closed points, and writing the domain and range using compound inequalities. Suitable for students in Grades 9-11.

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