Determine whether each relation defines a function, and give the domain and range.

검증된 단계별 안내

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.

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.

Step 3: Conclude that the relation defines a function because it passes the vertical line test.

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 \((-\infty, \infty)\).

Step 5: Determine the range by looking at the y-values covered by the graph. Similarly, since the line extends indefinitely in both directions along the y-axis, the range is all real numbers, expressed as \((-\infty, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means no x-value can be paired with more than one y-value. The vertical line test is a common method to determine if a graph represents 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 line extending infinitely in both directions, the domain is typically all real numbers, unless restricted by the context or graph.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. For a line with positive slope extending infinitely, the range is also all real numbers, as y-values increase and decrease without bound.