Determine Whether Each Of The Following Relationships Is A F

Determine whether each of the following relationships is a function and give the domain and range for each relation

Determine whether each of the following relationships is a function and give the domain and range for each relation. Relation:

(a) {(1,4), (4,4), (1,3), (6,3)}

(b) {(1,3), (2,5), (5,6), (3,7)}

(c) {(−3,−3), (−1,−1), (7,0)}

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In this analysis, we assess whether each given relation qualifies as a function and determine their respective domains and ranges. A relation is considered a function if each element in the domain corresponds to exactly one element in the range, meaning no input is associated with multiple outputs. Conversely, relations where a single input maps to multiple outputs violate the definition of a function.

Analysis of Relation (a): {(1,4), (4,4), (1,3), (6,3)}

Examining Relation (a), we observe that the element '1' in the domain maps to two different elements in the range: 4 and 3. Since an input value ('1') is associated with more than one output (4 and 3), this relation is not a function.

Domains and ranges for this relation:

  • Domain: {1, 4, 6}
  • Range: {3, 4}

Analysis of Relation (b): {(1,3), (2,5), (5,6), (3,7)}

Relation (b) demonstrates each particular input in the domain mapping to a unique output. For example, 1 maps to 3, 2 to 5, 5 to 6, and 3 to 7, with no input sharing multiple outputs. Therefore, this relation qualifies as a function.

Domains and ranges for this relation:

  • Domain: {1, 2, 3, 5}
  • Range: {3, 5, 6, 7}

Analysis of Relation (c): {(−3,−3), (−1,−1), (7,0)}

Relation (c) associates each input with a single output without any conflicts. The inputs -3, -1, and 7 each map to -3, -1, and 0 respectively, confirming it as a function.

Domains and ranges for this relation:

  • Domain: {−3, −1, 7}
  • Range: {−3, −1, 0}

Summary

Relations (b) and (c) are functions because each input maps to a single output. Relation (a) is not a function because the input '1' maps to multiple outputs, violating the function definition. Understanding whether a relation is a function is fundamental in mathematical analysis, as it influences how the relation can be graphed, analyzed, and applied in various contexts.

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