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: Determine if y is a function of x by using the vertical line test. If any vertical line intersects the graph at more than one point, then y is not a function of x.
2. Step 2: Observe the graph carefully. Notice that for each x-value between -10 and 10, there is exactly one y-value on the graph. Outside this interval, the graph consists of two separate branches that do not overlap vertically.
3. Step 3: Conclude that y is a function of x because the graph passes the vertical line test; no vertical line intersects the graph at more than one point.
4. Step 4: Identify the domain of the function. The graph extends horizontally from x = -10 to x = 10, so the domain is \([-10, 10]\).
5. Step 5: Identify the range of the function. The lowest y-value on the graph is 0 (at x = -10 and x = 10), and the graph extends upward without bound, so the range is \([0, \infty)\).

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 can intersect 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 is defined, while the range is the set of all possible y-values the function can take. Identifying these helps describe the extent of the relation.

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.