Which relation is a function?

 

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A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.
Image with alt text: A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin.

A coordinate grid with the graph of a circle centered at the origin and passing through the point begin ordered pair 2 comma 1 end ordered pair.
Image with alt text: A coordinate grid with the graph of a circle centered at the origin and passing through the point begin ordered pair 2 comma 1 end ordered pair.

Coordinate grid with graph of a vertical line at x equals 3.
Image with alt text: Coordinate grid with graph of a vertical line at x equals 3.

A coordinate grid containing a U shaped graph with arrows on its ends.  The U shape opens upwards and the bottom of the U shape is located at the origin.

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!

Which relation is a function? A coordinate grid containing a U shape with arrows on both ends that opens to the right. The bottom portion of the U passes through the origin. A coordinate grid with the graph of a circle centered at the origin and passing through the point (2,1). Coordinate grid with graph of a vertical line at x equals 3. A coordinate grid containing a U shaped graph with arrows on its ends. The U shape opens upwards and the bottom of the U shape is located at the origin.

Learn how to determine which relation is a function using the Vertical Line Test. Explore examples including a U-shaped graph, a circle centered at the origin, and a vertical line at x = 3. Suitable for high school students in Grades 9-12.

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