In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(3, −2), (5, −2), (7, 1), (4, 9)}

검증된 단계별 안내

Step 1: Recall the definition of a function. A relation is a function if each input (x-value) is paired with exactly one output (y-value). In other words, no x-value can be repeated with different y-values.

Step 2: Examine the given relation: {(3, −2), (5, −2), (7, 1), (4, 9)}. Identify the x-values (inputs) and check if any of them are repeated. The x-values are {3, 5, 7, 4}. Since none of these x-values are repeated, the relation is a function.

Step 3: Determine the domain of the relation. The domain is the set of all x-values (inputs) in the relation. From the given relation, the domain is {3, 5, 7, 4}.

Step 4: Determine the range of the relation. The range is the set of all y-values (outputs) in the relation. From the given relation, the y-values are {−2, −2, 1, 9}. Since we list each value only once, the range is {−2, 1, 9}.

Step 5: Summarize the findings. The relation is a function because no x-value is repeated with different y-values. The domain is {3, 5, 7, 4}, and the range is {−2, 1, 9}.

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주요 개념

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Function Definition

A function is a specific type of relation where each input (or domain element) is associated with exactly one output (or range element). This means that no two ordered pairs can have the same first element with different second elements. Understanding this definition is crucial for determining if a given relation qualifies as a function.

Domain and Range

The domain of a relation is the set of all possible input values (first elements of the ordered pairs), while the range is the set of all possible output values (second elements). Identifying the domain and range helps in understanding the behavior of the function and the values it can take. For the given relation, extracting these sets is essential for a complete analysis.

Ordered Pairs

Ordered pairs are pairs of numbers written in the form (x, y), where 'x' represents the input and 'y' represents the output. In the context of relations and functions, the order of these pairs is significant, as it determines the relationship between the input and output. Analyzing the ordered pairs in the given relation is necessary to assess whether it meets the criteria of a function.