Question

select is a function or is not a function to correctly classify each relation.
{ (2, 2), (4, 4), (6, 6), (8, 8) } is a function is not a function
{ (0, 3), (3, 5), (5, 6), (8, 4) } is a function is not a function
{ (1, 2), (3, 3), (4, 8), (6, 3) } is a function is not a function
{ (3, 4), (5, 2), (5, 6), (7, 3) } is a function is not a function

Explanation:

To determine if a relation is a function, we use the definition: a relation is a function if every input (first element of each ordered pair) has exactly one output (second element of each ordered pair). In other words, no two ordered pairs can have the same first element with different second elements.

For the relation \(\{(2, 2), (4, 4), (6, 6), (8, 8)\}\)

• Check the first elements: \(2, 4, 6, 8\).
• Each first element appears only once, so each input has exactly one output.
• Classification: Is a Function

For the relation \(\{(0, 3), (3, 5), (5, 6), (8, 4)\}\)

• Check the first elements: \(0, 3, 5, 8\).
• Each first element appears only once, so each input has exactly one output.
• Classification: Is a Function

For the relation \(\{(1, 2), (3, 3), (4, 8), (6, 3)\}\)

• Check the first elements: \(1, 3, 4, 6\).
• Each first element appears only once (even though \(3\) is the output for two different inputs, the inputs themselves are unique).
• Classification: Is a Function

For the relation \(\{(3, 4), (5, 2), (5, 6), (7, 3)\}\)

• Check the first elements: \(3, 5, 5, 7\).
• The input \(5\) appears twice with different outputs (\(2\) and \(6\)).
• Classification: Is not a Function

Final Classifications:

Relation	Is a Function	Is not a Function
\(\{(0, 3), (3, 5), (5, 6), (8, 4)\}\)	✔️
\(\{(1, 2), (3, 3), (4, 8), (6, 3)\}\)	✔️
\(\{(3, 4), (5, 2), (5, 6), (7, 3)\}\)		✔️

Answer:

To determine if a relation is a function, we use the definition: a relation is a function if every input (first element of each ordered pair) has exactly one output (second element of each ordered pair). In other words, no two ordered pairs can have the same first element with different second elements.

For the relation \(\{(2, 2), (4, 4), (6, 6), (8, 8)\}\)

• Check the first elements: \(2, 4, 6, 8\).
• Each first element appears only once, so each input has exactly one output.
• Classification: Is a Function

For the relation \(\{(0, 3), (3, 5), (5, 6), (8, 4)\}\)

• Check the first elements: \(0, 3, 5, 8\).
• Each first element appears only once, so each input has exactly one output.
• Classification: Is a Function

For the relation \(\{(1, 2), (3, 3), (4, 8), (6, 3)\}\)

• Check the first elements: \(1, 3, 4, 6\).
• Each first element appears only once (even though \(3\) is the output for two different inputs, the inputs themselves are unique).
• Classification: Is a Function

For the relation \(\{(3, 4), (5, 2), (5, 6), (7, 3)\}\)

• Check the first elements: \(3, 5, 5, 7\).
• The input \(5\) appears twice with different outputs (\(2\) and \(6\)).
• Classification: Is not a Function

Final Classifications:

Relation	Is a Function	Is not a Function
\(\{(0, 3), (3, 5), (5, 6), (8, 4)\}\)	✔️
\(\{(1, 2), (3, 3), (4, 8), (6, 3)\}\)	✔️
\(\{(3, 4), (5, 2), (5, 6), (7, 3)\}\)		✔️