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 graph type. The graph shows a circle centered at (20, 10). A circle is defined by the equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Step 2: Determine if \(y\) is a function of \(x\). For \(y\) to be a function of \(x\), each \(x\)-value must correspond to exactly one \(y\)-value. In a circle, for many \(x\)-values, there are two corresponding \(y\)-values (one above and one below the center), so \(y\) is not a function of \(x\).
3. Step 3: Find the domain of the relation. The domain is the set of all possible \(x\)-values on the circle. Since the circle is centered at \(x=20\) and extends equally in both directions by the radius, the domain is \([20 - r, 20 + r]\).
4. Step 4: Find the range of the relation. The range is the set of all possible \(y\)-values on the circle. Since the circle is centered at \(y=10\) and extends equally in both directions by the radius, the range is \([10 - r, 10 + r]\).
5. Step 5: Summarize: \(y\) is not a function of \(x\) because the vertical line test fails (vertical lines intersect the circle at two points). The domain and range are intervals centered at the circle's center coordinates, extending by the 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.

Domain and Range of a Relation

The domain is the set of all possible x-values in the relation, while the range is the set of all possible y-values. For a circle centered at (20, 10) with radius 10, the domain and range are intervals around these values.