Determine whether each relation defines a function, and give the domain and range. {(1,1),(1,-1),(0,0),(2,4),(2,-4)}

검증된 단계별 안내

Recall that a relation defines a function if every input (x-value) corresponds to exactly one output (y-value).

Examine the given set of ordered pairs: \(\{(1,1), (1,-1), (0,0), (2,4), (2,-4)\}\). Identify the x-values: 1, 1, 0, 2, 2.

Notice that the x-values 1 and 2 each appear more than once but are paired with different y-values (1 and -1 for x=1; 4 and -4 for x=2). This means the relation does not assign a unique output for these inputs.

Conclude that this relation does not define a function because some inputs have multiple outputs.

To find the domain, list all unique x-values: \(\{0, 1, 2\}\). To find the range, list all unique y-values: \(\{1, -1, 0, 4, -4\}\).

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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 (or domain element) corresponds to exactly one output (or range element). If any input is paired with more than one output, the relation is not a function. This concept helps determine if the given set of ordered pairs qualifies as a function.

Domain of a Relation

The domain is the set of all possible input values (first elements) in the relation. Identifying the domain involves listing all unique x-values from the ordered pairs. Understanding the domain is essential for describing the inputs over which the relation or function is defined.

Range of a Relation

The range is the set of all possible output values (second elements) in the relation. It includes all unique y-values from the ordered pairs. Knowing the range helps describe the outputs that the relation or function can produce.