The question asks if the given graph represents a function of $x$, and to provide a reason.

To determine if a graph is a function, we use the vertical line test. According to this test, if any vertical line drawn on the graph intersects it at more than one point, the graph is not a function of $x$. This is because, for a function, each input $x$ must map to exactly one output $y$.

In the graph you've provided, there is a part of the curve where a vertical line would intersect the graph at more than one point (specifically, around the middle portion where the curve loops back on itself). Therefore, this graph is not a function of $x$.

Reason:

Since a vertical line can intersect the graph at multiple points, it fails the vertical line test, meaning that for some values of $x$, there are multiple values of $y$. Hence, it is not a function.

Would you like more details or have any further questions?

Here are five related questions to consider:

1. What is the formal definition of a function?
2. How can we test if a relation is a function algebraically, aside from using the vertical line test?
3. What are some common examples of graphs that fail the vertical line test?
4. How does the concept of a "one-to-one" function differ from just a "function"?
5. Can the graph of a circle represent a function? Why or why not?

Tip:

When analyzing graphs to determine if they represent functions, always visualize or apply the vertical line test. If the line touches more than once, it's not a function!