Determine whether each relation defines a function, and give the domain and range.

Verified step by step guidance

1. Step 1: Identify the relation shown on the graph. The graph depicts an oval shape, which is a closed curve representing a set of points (x, y).
2. Step 2: Determine if the relation defines a function by using the Vertical Line Test. This test states that if any vertical line intersects the graph at more than one point, the relation is not a function.
3. Step 3: Apply the Vertical Line Test to the oval. Notice that vertical lines passing through the middle of the oval intersect the graph at two points, meaning for some x-values there are two corresponding y-values.
4. Step 4: Conclude that the relation is not a function because it fails the Vertical Line Test (some x-values correspond to multiple y-values).
5. Step 5: Find the domain and range by looking at the extent of the oval along the x-axis and y-axis. The domain is the set of all x-values covered by the oval, and the range is the set of all y-values covered by the oval.

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means no vertical line intersects the graph at more than one point, ensuring each x has a unique y.

Domain of a Relation

The domain is the set of all possible input values (x-values) for the relation. It includes every x-coordinate that appears on the graph, representing all values for which the relation is defined.

Range of a Relation

The range is the set of all possible output values (y-values) for the relation. It includes every y-coordinate that appears on the graph, representing all values the relation can produce.