Let's determine which ordered pairs prevent the given relation [tex]\( R \)[/tex] from being a function. Recall that for a relation to be a function, each input (first element of the ordered pair) must be associated with exactly one output (second element of the ordered pair).
Given the relation as a set of ordered pairs:
[tex]\[ R = \{ (1, 4), (1, 3), (-1, 3), (2, 15) \} \][/tex]
To see which pairs prevent the relation from being a function, we need to identify if there are any repeated [tex]\( x \)[/tex]-values with different [tex]\( y \)[/tex]-values.
### Step-by-Step Solution
1. Identify pairs with the same x-value:
- Look at the first element (x-value) of each ordered pair.
- Check if any [tex]\( x \)[/tex]-value appears more than once with different [tex]\( y \)[/tex]-values.
Analyzing the given pairs:
- [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] both have the same [tex]\( x \)[/tex]-value of [tex]\( 1 \)[/tex], but different [tex]\( y \)[/tex]-values of [tex]\( 4 \)[/tex] and [tex]\( 3 \)[/tex].
Thus, [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] prevent the relation from being a function because they violate the requirement that each [tex]\( x \)[/tex]-value must map to exactly one [tex]\( y \)[/tex]-value. This means the relation is not a function.
2. Identify pairs with the same y-value:
- Now, look at the second element (y-value) of each ordered pair.
- Check if any [tex]\( y \)[/tex]-value appears more than once with different [tex]\( x \)[/tex]-values.
Analyzing the given pairs:
- [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] both have the same [tex]\( y \)[/tex]-value of [tex]\( 3 \)[/tex], but different [tex]\( x \)[/tex]-values of [tex]\( 1 \)[/tex] and [tex]\(-1 \)[/tex].
While this does not prevent the relation from being a function, this information can be useful for understanding the structure of the relation.
### Conclusion
The ordered pairs that prevent this relation from being a function due to having the same [tex]\( x \)[/tex]-value but different [tex]\( y \)[/tex]-values are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
Additionally, pairs with the same [tex]\( y \)[/tex]-value but different [tex]\( x \)[/tex]-values are:
[tex]\[ (1, 3) \quad \text{and} \quad (-1, 3) \][/tex]
However, the key pairs that directly impact the relation from being a function are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
