Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Verified step by step guidance

1. Step 1: Identify if the relation is a function 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 of the red line. Since it is a straight line with a positive slope and no vertical segments, any vertical line will intersect it exactly once.
3. Step 3: Conclude that the relation is a function because it passes the vertical line test.
4. Step 4: Determine the domain by looking at the x-values covered by the graph. Since the line extends indefinitely in both directions along the x-axis, the domain is all real numbers, expressed as \((-\infty, \infty)\).
5. Step 5: Determine the range by looking at the y-values covered by the graph. Because the line extends indefinitely in both directions along the y-axis, the range is also all real numbers, expressed as \((-\infty, \infty)\).

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 more than once, ensuring each x has a unique y. Checking this helps determine if the relation is a function.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For a graph, it includes all x-values covered by the graph. Understanding the domain helps identify the valid inputs for the function.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. On a graph, it includes all y-values the graph attains. Knowing the range helps understand the possible outputs of the function.