To determine which relation is a function, we need to apply the Vertical Line Test. A graph represents a function if, and only if, no vertical line intersects the graph at more than one point.

Now, let's analyze each of the given graphs:

1. U shape with arrows on both ends that opens to the right:

   • This graph does not pass the vertical line test because a vertical line will intersect the U-shape more than once.
   • Not a function.

2. A circle centered at the origin passing through (2,1):

   • A vertical line would intersect the circle at two points for some values of $x$.
   • Not a function.

3. Vertical line at $x = 3$:

   • A vertical line itself fails the vertical line test because it is a vertical line and intersects itself infinitely.
   • Not a function.

4. U shape opening upwards with its vertex at the origin:

   • This graph passes the vertical line test. No vertical line will intersect the U-shaped curve more than once.
   • This is a function.

Conclusion:

The U shape that opens upwards with its vertex at the origin is a function.

Would you like more details on any specific graph or concept?

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Here are 5 related questions:

1. How does the vertical line test work in determining functions?
2. Can a graph represent a function if it intersects the vertical line at multiple points?
3. Why is the graph of a circle not a function?
4. What is the difference between a function and a relation in mathematics?
5. How do different transformations affect whether a graph is a function?

Tip: When graphing functions, always check for multiple intersections with a vertical line to confirm the function status!