Answer :
Certainly! Let's tackle each part of the question step-by-step.
### Part (a): Determining if the Given Relation is a Function
A relation is considered a function if every input (or [tex]\( x \)[/tex]-value) corresponds to exactly one output (or [tex]\( y \)[/tex]-value). This means that no [tex]\( x \)[/tex]-value should be repeated with different [tex]\( y \)[/tex]-values.
Given the relation [tex]\(\{(2,6),(-5,1),(-7,4)\}\)[/tex], let's list the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values:
- [tex]\( x = 2 \)[/tex] corresponds to [tex]\( y = 6 \)[/tex]
- [tex]\( x = -5 \)[/tex] corresponds to [tex]\( y = 1 \)[/tex]
- [tex]\( x = -7 \)[/tex] corresponds to [tex]\( y = 4 \)[/tex]
Now, check if any [tex]\( x \)[/tex]-values are repeated:
- [tex]\( x = 2 \)[/tex] appears once
- [tex]\( x = -5 \)[/tex] appears once
- [tex]\( x = -7 \)[/tex] appears once
Since each [tex]\( x \)[/tex]-value appears only once, there are no [tex]\( x \)[/tex]-values that are repeated. Therefore, the given relation is a function.
### Part (b): Adding a Fourth Point to Make the Relation Not a Function
To make a relation not a function, we need to add a new point where the [tex]\( x \)[/tex]-value is already present in the relation but corresponds to a different [tex]\( y \)[/tex]-value. This way, an [tex]\( x \)[/tex]-value will be mapped to more than one [tex]\( y \)[/tex]-value.
Let's consider the existing [tex]\( x \)[/tex]-values: [tex]\( 2, -5, \)[/tex] and [tex]\( -7 \)[/tex]. We can choose any of these [tex]\( x \)[/tex]-values and pair it with a different [tex]\( y \)[/tex]-value.
For instance, adding the point [tex]\( (2, 8) \)[/tex] will work:
- The point [tex]\( (2, 6) \)[/tex] is already in the relation.
- Adding [tex]\( (2, 8) \)[/tex] introduces another pair with the same [tex]\( x \)[/tex]-value but a different [tex]\( y \)[/tex]-value.
The updated relation becomes:
[tex]\[ \{(2,6),(-5,1),(-7,4),(2,8)\} \][/tex]
Now, the [tex]\( x \)[/tex]-value [tex]\( 2 \)[/tex] is associated with two different [tex]\( y \)[/tex]-values ([tex]\( 6 \)[/tex] and [tex]\( 8 \)[/tex]). This makes the updated relation not a function.
### Summary
- (a) The given relation [tex]\(\{(2,6),(-5,1),(-7,4)\}\)[/tex] is a function because every [tex]\( x \)[/tex]-value is unique.
- (b) By adding the point [tex]\((2, 8)\)[/tex], the updated relation becomes [tex]\(\{(2,6),(-5,1),(-7,4),(2,8)\}\)[/tex], which is not a function because the [tex]\( x \)[/tex]-value [tex]\(2\)[/tex] corresponds to more than one [tex]\( y \)[/tex]-value.
This completes the step-by-step solution for the question.
### Part (a): Determining if the Given Relation is a Function
A relation is considered a function if every input (or [tex]\( x \)[/tex]-value) corresponds to exactly one output (or [tex]\( y \)[/tex]-value). This means that no [tex]\( x \)[/tex]-value should be repeated with different [tex]\( y \)[/tex]-values.
Given the relation [tex]\(\{(2,6),(-5,1),(-7,4)\}\)[/tex], let's list the [tex]\( x \)[/tex]-values and [tex]\( y \)[/tex]-values:
- [tex]\( x = 2 \)[/tex] corresponds to [tex]\( y = 6 \)[/tex]
- [tex]\( x = -5 \)[/tex] corresponds to [tex]\( y = 1 \)[/tex]
- [tex]\( x = -7 \)[/tex] corresponds to [tex]\( y = 4 \)[/tex]
Now, check if any [tex]\( x \)[/tex]-values are repeated:
- [tex]\( x = 2 \)[/tex] appears once
- [tex]\( x = -5 \)[/tex] appears once
- [tex]\( x = -7 \)[/tex] appears once
Since each [tex]\( x \)[/tex]-value appears only once, there are no [tex]\( x \)[/tex]-values that are repeated. Therefore, the given relation is a function.
### Part (b): Adding a Fourth Point to Make the Relation Not a Function
To make a relation not a function, we need to add a new point where the [tex]\( x \)[/tex]-value is already present in the relation but corresponds to a different [tex]\( y \)[/tex]-value. This way, an [tex]\( x \)[/tex]-value will be mapped to more than one [tex]\( y \)[/tex]-value.
Let's consider the existing [tex]\( x \)[/tex]-values: [tex]\( 2, -5, \)[/tex] and [tex]\( -7 \)[/tex]. We can choose any of these [tex]\( x \)[/tex]-values and pair it with a different [tex]\( y \)[/tex]-value.
For instance, adding the point [tex]\( (2, 8) \)[/tex] will work:
- The point [tex]\( (2, 6) \)[/tex] is already in the relation.
- Adding [tex]\( (2, 8) \)[/tex] introduces another pair with the same [tex]\( x \)[/tex]-value but a different [tex]\( y \)[/tex]-value.
The updated relation becomes:
[tex]\[ \{(2,6),(-5,1),(-7,4),(2,8)\} \][/tex]
Now, the [tex]\( x \)[/tex]-value [tex]\( 2 \)[/tex] is associated with two different [tex]\( y \)[/tex]-values ([tex]\( 6 \)[/tex] and [tex]\( 8 \)[/tex]). This makes the updated relation not a function.
### Summary
- (a) The given relation [tex]\(\{(2,6),(-5,1),(-7,4)\}\)[/tex] is a function because every [tex]\( x \)[/tex]-value is unique.
- (b) By adding the point [tex]\((2, 8)\)[/tex], the updated relation becomes [tex]\(\{(2,6),(-5,1),(-7,4),(2,8)\}\)[/tex], which is not a function because the [tex]\( x \)[/tex]-value [tex]\(2\)[/tex] corresponds to more than one [tex]\( y \)[/tex]-value.
This completes the step-by-step solution for the question.
