For each graph, determine whether y is a function of x. Give the domain and range of each relation.

Verified step by step guidance

1. Step 1: Identify the shape and its center. The graph shows a circle centered at the point (20, 10).
2. Step 2: Recall the definition of a function. A relation is a function if for every x-value there is exactly one y-value.
3. Step 3: Use the vertical line test to determine if y is a function of x. If any vertical line intersects the graph at more than one point, then y is not a function of x.
4. Step 4: Apply the vertical line test to the circle. Since a vertical line passing through the circle will intersect it at two points (except at the extreme edges), y is not a function of x.
5. Step 5: Determine the domain and range. The domain is the set of all x-values covered by the circle, which extends from (20 - radius) to (20 + radius). The range is the set of all y-values covered by the circle, which extends from (10 - radius) to (10 + radius).

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.

Vertical Line Test

The vertical line test is a visual method to determine if a graph represents a function. If any vertical line crosses the graph more than once, the graph does not represent a function because an x-value has multiple y-values.

Domain and Range of a Circle

The domain of a circle includes all x-values covered by the circle, and the range includes all y-values covered. For a circle centered at (h, k) with radius r, domain is [h-r, h+r] and range is [k-r, k+r].