To determine whether the graphed relation defines a function, we need to apply the vertical line test. According to this test, a graph represents a function if and only if every vertical line drawn through the graph intersects it at most once.

Analysis of the Graph:

The graph provided has a wave-like curve. When you apply the vertical line test:

• If any vertical line drawn intersects the curve more than once, the relation is not a function.
• If every vertical line intersects the curve only once, then the relation is a function.

In this graph, you can see that there are multiple points where vertical lines would intersect the graph more than once. For example, if you draw a vertical line at $x = 1$, it would intersect the graph in two places, meaning that for the same $x$-value, there are two different $y$-values.

Conclusion:

Since there are vertical lines that intersect the graph at more than one point, the graphed relation does not define a function.

Domain and Range:

• Domain: The graph spans from approximately $x = -10$ to $x = 10$, so the domain can be written as $[-10, 10]$.
• Range: The graph ranges from approximately $y = -10$ to $y = 10$, so the range can be written as $[-10, 10]$.

Would you like more details on this analysis, or do you have any questions?

Here are some related questions that might expand on this topic:

1. What is the vertical line test and how is it applied in different types of graphs?
2. How do we find the domain and range for a relation on a graph?
3. What are some examples of relations that are not functions?
4. Can a function have multiple $y$-values for the same $x$-value? Why or why not?
5. How can we determine if a function is one-to-one using a horizontal line test?

Tip: Always remember that a function will pass the vertical line test, which means no vertical line can intersect the graph more than once at any point.