To determine whether a relation is a function, we need to check that each [tex]\( x \)[/tex]-value in the ordered pairs [tex]\( (x, y) \)[/tex] maps to exactly one [tex]\( y \)[/tex]-value. That is, no [tex]\( x \)[/tex]-value should be repeated with a different [tex]\( y \)[/tex].
Given the relation:
[tex]\[
R = \{(1,4), (1,3), (-1,3), (2,15)\}
\][/tex]
Let's analyze each step carefully:
1. List the ordered pairs and identify [tex]\( x \)[/tex]-values:
[tex]\[
\begin{aligned}
&\text{Pair } (1, 4) \to x = 1, \\
&\text{Pair } (1, 3) \to x = 1, \\
&\text{Pair } (-1, 3) \to x = -1, \\
&\text{Pair } (2, 15) \to x = 2.
\end{aligned}
\][/tex]
2. Check for repeated [tex]\( x \)[/tex]-values:
- The [tex]\( x \)[/tex]-value 1 appears in two different ordered pairs: [tex]\((1,4)\)[/tex] and [tex]\((1, 3)\)[/tex].
- Therefore, the pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] have the same [tex]\( x \)[/tex]-value, which violates the definition of a function. There are no other [tex]\( x \)[/tex]-values that repeat with different [tex]\( y \)[/tex]-values.
3. Double-check other ordered pairs:
- The pair [tex]\((-1, 3)\)[/tex] has an [tex]\( x \)[/tex]-value of -1, only appears once.
- The pair [tex]\((2, 15)\)[/tex] has an [tex]\( x \)[/tex]-value of 2, only appears once.
Next, let's address the pairs that have the same [tex]\( y \)[/tex]-value:
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] both map to the [tex]\( y \)[/tex]-value of 3. This is allowed and does not prevent the relation from being a function since there is no restriction that distinct [tex]\( x \)[/tex]-values must have distinct [tex]\( y \)[/tex]-values in a function. The critical criterion is that same [tex]\( x \)[/tex]-values should not map to different [tex]\( y \)[/tex]-values.
Thus, the ordered pairs that prevent this relation from being a function are:
[tex]\[
(1, 4) \quad \text{and} \quad (1, 3)
\][/tex]
Also, reviewing the problematic pair and ensuring correctness:
- The ordered pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] prevent the relation from being a function because they have the same [tex]\( x \)[/tex]-value.
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] having the same [tex]\( y \)[/tex]-value do not affect the definition of a function.
Conclusively, the correct pairs preventing the relation from being a function are:
[tex]\[
(1, 4) \quad \text{and} \quad (1, 3)
\][/tex]
