Question

question
each answer choice below represents a relation by a set of ordered pairs. in which of the answer choices is the relation a function?
select all correct answers.
select all that apply:
{\\( (2,1),(1,3),(-5,9),(8,1) \\)}
{\\( (2,1),(-2,-1),(2,12),(4,7) \\)}
{\\( (9,-2),(4,13),(0,7),(3,13) \\)}
{\\( (7,-2),(2,9),(-1,6),(1,6) \\)}
{\\( (-5,10),(6,9),(7,-2),(6,5) \\)}

Explanation:

To determine if a relation (set of ordered pairs) is a function, we use the definition: a function is a relation where each input (first element of the ordered pair, the domain) has exactly one output (second element, the range). So we check if any x - value (first coordinate) is repeated with different y - values.

Step 1: Analyze the first set \(\{(2,1),(1,3),(-5,9),(8,1)\}\)

• The x - values are \(2\), \(1\), \(-5\), \(8\).
• Each x - value appears only once. So this relation is a function.

Step 2: Analyze the second set \(\{(2,1),(-2,-1),(2,12),(4,7)\}\)

• The x - value \(2\) appears twice, with \(y\) - values \(1\) and \(12\).
• Since one input (\(2\)) has two different outputs, this relation is not a function.

Step 3: Analyze the third set \(\{(9,-2),(4,13),(0,7),(3,13)\}\)

• The x - values are \(9\), \(4\), \(0\), \(3\).
• Each x - value appears only once (note that \(13\) is a repeated \(y\) - value, but we only care about repeated \(x\) - values with different \(y\) - values). So this relation is a function.

Step 4: Analyze the fourth set \(\{(7,-2),(2,9),(-1,6),(1,6)\}\)

• The x - values are \(7\), \(2\), \(-1\), \(1\).
• Each x - value appears only once (note that \(6\) is a repeated \(y\) - value, but we only care about repeated \(x\) - values with different \(y\) - values). So this relation is a function.

Step 5: Analyze the fifth set \(\{(-5,10),(6,9),(7,-2),(6,5)\}\)

• The x - value \(6\) appears twice, with \(y\) - values \(9\) and \(5\).
• Since one input (\(6\)) has two different outputs, this relation is not a function.

Answer:

The relations that are functions are:

• \(\{(2,1),(1,3),(-5,9),(8,1)\}\)
• \(\{(9,-2),(4,13),(0,7),(3,13)\}\)
• \(\{(7,-2),(2,9),(-1,6),(1,6)\}\)