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 if y is a function of x by using the vertical line test. A relation is a function if every vertical line intersects the graph at most once.
2. Step 2: Observe the graph carefully. Notice that at x = 0, there are two different y-values: y = 2 (solid point) and y = 4 (open circle). This means a vertical line at x = 0 intersects the graph twice.
3. Step 3: Since the vertical line at x = 0 intersects the graph more than once, y is not a function of x for this relation.
4. Step 4: Determine the domain by looking at the x-values covered by the graph. The graph extends from about x = -20 to x = 20, so the domain is approximately \([-20, 20]\).
5. Step 5: Determine the range by looking at the y-values covered by the graph. The lowest y-value is about 2 (solid point at x=0), and the graph extends upward beyond y = 10, so the range is approximately \([2, \infty)\).

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 and Range

The domain is the set of all possible x-values for which the function or relation is defined. The range is the set of all possible y-values that the function or relation can take. Identifying these helps describe the extent of the graph.

Vertical Line Test

The vertical line test is a visual way 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 maps to multiple y-values.