Question

1. the table shows two relations, a and b. which relation is not a function? which single value of x or y could be changed to make the relation a function? justify your answers.

relation a table:

x	y
-4	5
0	-4
-4	-2

relation b table:

x	y
-3	-3
3	3
0	3

Explanation:

Step1: Recall Function Definition

A function is a relation where each input (x - value) has exactly one output (y - value). So we check relations A and B for this property.

Step2: Analyze Relation A

In relation A, the x - value \(-4\) is paired with \(y = 5\) and \(y=-2\). Since one input (\(-4\)) has two different outputs, relation A is not a function.

Step3: Analyze Relation B

In relation B, each x - value (\(-3\), \(3\), \(0\)) has a unique y - value (\(-3\), \(3\), \(3\) respectively). So relation B is a function.

Step4: Fixing Relation A

To make relation A a function, we can change either the x - value or the y - value of one of the pairs with \(x=-4\). For example, we can change the \(x\) - value of the pair \((-4,-2)\) to a new value (not \(-4\) or \(0\), say \(1\)), or change the \(y\) - value of one of the \(-4\) pairs. If we change the second \(-4\) (in the pair \((-4,-2)\)) to a different x - value, like \(1\), then each x - value will have one y - value. Or we can change one of the y - values for \(x = - 4\), say change \(5\) to \(6\) or change \(-2\) to \(7\), so that \(x=-4\) has only one y - value.

Answer:

Relation A is not a function because the input \(x = - 4\) has two different outputs (\(y = 5\) and \(y=-2\)). To make relation A a function, we can change the \(x\) - value of one of the pairs with \(x=-4\) (e.g., change the \(x\) in \((-4,-2)\) to a new value not equal to \(-4\) or \(0\)) or change one of the \(y\) - values for \(x=-4\) (e.g., change \(5\) to a different number or change \(-2\) to a different number) so that each \(x\) - value has exactly one \(y\) - value.