Answer :
To determine which value of [tex]\( x \)[/tex] will make the given relation [tex]\( \{(0,4), (2,6), (4,8), (x,7)\} \)[/tex] a function, we need to understand that in a function, each [tex]\( x \)[/tex]-value must be unique. This means that no two ordered pairs can have the same [tex]\( x \)[/tex] value.
Let's examine the options for [tex]\( x \)[/tex]:
1. If [tex]\( x = 0 \)[/tex]:
- The ordered pair would be [tex]\( (0,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (0,4) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 0 is repeated, [tex]\( x = 0 \)[/tex] would not make the relation a function.
2. If [tex]\( x = 2 \)[/tex]:
- The ordered pair would be [tex]\( (2,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (2,6) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 2 is repeated, [tex]\( x = 2 \)[/tex] would not make the relation a function.
3. If [tex]\( x = 6 \)[/tex]:
- The ordered pair would be [tex]\( (6,7) \)[/tex].
- There are no other ordered pairs with an [tex]\( x \)[/tex]-value of 6.
- Therefore, [tex]\( x = 6 \)[/tex] would keep all [tex]\( x \)[/tex]-values unique, making the relation a function.
4. If [tex]\( x = 4 \)[/tex]:
- The ordered pair would be [tex]\( (4,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (4,8) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 4 is repeated, [tex]\( x = 4 \)[/tex] would not make the relation a function.
So, the only value of [tex]\( x \)[/tex] that would make the given relation a function is [tex]\( x = 6 \)[/tex]. Therefore, the correct answer is:
(3) 6
Let's examine the options for [tex]\( x \)[/tex]:
1. If [tex]\( x = 0 \)[/tex]:
- The ordered pair would be [tex]\( (0,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (0,4) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 0 is repeated, [tex]\( x = 0 \)[/tex] would not make the relation a function.
2. If [tex]\( x = 2 \)[/tex]:
- The ordered pair would be [tex]\( (2,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (2,6) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 2 is repeated, [tex]\( x = 2 \)[/tex] would not make the relation a function.
3. If [tex]\( x = 6 \)[/tex]:
- The ordered pair would be [tex]\( (6,7) \)[/tex].
- There are no other ordered pairs with an [tex]\( x \)[/tex]-value of 6.
- Therefore, [tex]\( x = 6 \)[/tex] would keep all [tex]\( x \)[/tex]-values unique, making the relation a function.
4. If [tex]\( x = 4 \)[/tex]:
- The ordered pair would be [tex]\( (4,7) \)[/tex].
- However, there is already an ordered pair [tex]\( (4,8) \)[/tex].
- Since the [tex]\( x \)[/tex]-value 4 is repeated, [tex]\( x = 4 \)[/tex] would not make the relation a function.
So, the only value of [tex]\( x \)[/tex] that would make the given relation a function is [tex]\( x = 6 \)[/tex]. Therefore, the correct answer is:
(3) 6
