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