https://file.aiquickdraw.com/mathbot/file/1729628378890r41ystof.jpg

Let's analyze the two graphs to determine if each represents a function of \( x \). To do this, we can apply the **vertical line test**, which states that if any vertical line drawn on the graph intersects the curve more than once, the curve is not a function of \( x \).

### Graph I:
- In this graph, every vertical line drawn on the curve will intersect the graph at only one point. This satisfies the vertical line test.
- Therefore, **Graph I** is the graph of a function of \( x \).

#### Domain and Range of Graph I:
- **Domain**: The \( x \)-values start from \( 0 \) and go up to \( 3 \), so the domain is \( [0, 3] \).
- **Range**: The \( y \)-values start from \( 0 \) and reach up to \( 2 \), so the range is \( [0, 2] \).

### Graph II:
- In this graph, there are vertical lines (like \( x = 0.5 \)) that intersect the graph at two points, which violates the vertical line test.
- Therefore, **Graph II** is **not** the graph of a function of \( x \).

Let me know if you need further clarifications or details!

### Follow-up questions:
1. What is the significance of the vertical line test in determining functions?
2. Could you identify a case where a curve would not pass the vertical line test?
3. What happens if the graph represents a function that is undefined for some values of \( x \)?
4. How can the domain and range change if the curve is extended?
5. What type of function does Graph I likely represent based on its shape?

### Tip:
For checking whether a graph is a function, always consider if each input (x-value) corresponds to exactly one output (y-value).

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Explore how to determine if a curve represents a function of x using the vertical line test. This problem also covers how to find the domain and range of functions, with detailed examples. Suitable for high school students in Grades 9-12.

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