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find the domain and range using inequality notation

The problem asks for the domain and range of the function shown in the graph, using inequality notation.

### Step 1: Determine the Domain
The domain refers to the set of all possible input values (x-values) that the function can take. From the graph:

- The function appears to start at \( x = -2 \) and end at \( x = 3 \).
- The function is defined for every value of \( x \) between \( -2 \) and \( 3 \), inclusive.

Thus, the domain in inequality notation is:
\[
-2 \leq x \leq 3
\]

### Step 2: Determine the Range
The range refers to the set of all possible output values (y-values) that the function can produce. From the graph:

- The lowest point on the graph appears to be \( y = -1 \).
- The highest point on the graph appears to be \( y = 2 \).

Therefore, the range in inequality notation is:
\[
-1 \leq y \leq 2
\]

### Final Answer:
- **Domain:** \( -2 \leq x \leq 3 \)
- **Range:** \( -1 \leq y \leq 2 \)

Would you like more details, or do you have any further questions?

Here are five related questions:
1. How would the domain and range change if the graph was shifted 2 units up?
2. What are the domain and range for a function that is defined for all real numbers?
3. How do you find the domain of a function that includes a square root?
4. What is the range of a quadratic function?
5. How does the range of a function change if you multiply the function by a negative number?

**Tip:** Always check both the minimum and maximum values on the graph when determining the range of a function.

Learn how to find the domain and range of a function using inequality notation. This problem involves analyzing a graph to determine the set of possible input and output values. Suitable for high school students.

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