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).