To determine which of the given options represents a function, we need to understand the definition of a function in terms of mappings from domain to range. A set of ordered pairs represents a function if each input (or [tex]\( x \)[/tex]-value) is associated with exactly one output (or [tex]\( y \)[/tex]-value). This means that no [tex]\( x \)[/tex]-value is repeated with a different [tex]\( y \)[/tex]-value.
Let's analyze the given options:
### Option B
The set of coordinates is: [tex]\(\{(-1,-11),(0,-7),(1,-3),(-1,5),(2,0)\}\)[/tex].
To check if this set of coordinates represents a function, let's list out the [tex]\( x \)[/tex]-values:
- From [tex]\((-1,-11)\)[/tex] : [tex]\( x = -1 \)[/tex]
- From [tex]\((0,-7)\)[/tex] : [tex]\( x = 0 \)[/tex]
- From [tex]\((1,-3)\)[/tex] : [tex]\( x = 1 \)[/tex]
- From [tex]\((-1,5)\)[/tex] : [tex]\( x = -1 \)[/tex]
- From [tex]\((2,0)\)[/tex] : [tex]\( x = 2 \)[/tex]
We observe that the [tex]\( x \)[/tex]-value [tex]\(-1\)[/tex] appears twice, once with the [tex]\( y \)[/tex]-value [tex]\(-11\)[/tex] and once with the [tex]\( y \)[/tex]-value [tex]\(5\)[/tex]. This means that for [tex]\( x = -1 \)[/tex], we have two different outputs ([tex]\(-11\)[/tex] and [tex]\(5\)[/tex]), indicating that option B does not represent a function.
### Option C
The table given is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -18 & -13 & 3 & 5 & -6 & 3 \\
\hline
y & -7 & -2 & 14 & 16 & 5 & 19 \\
\hline
\end{array}
\][/tex]
To check if this table represents a function, let's list out the [tex]\( x \)[/tex]-values and check for duplicates:
- [tex]\(-18\)[/tex]
- [tex]\(-13\)[/tex]
- [tex]\(3\)[/tex]
- [tex]\(5\)[/tex]
- [tex]\(-6\)[/tex]
- [tex]\(3\)[/tex]
Here, we observe that the [tex]\( x \)[/tex]-value [tex]\(3\)[/tex] appears twice, once with the [tex]\( y \)[/tex]-value [tex]\(14\)[/tex] and once with the [tex]\( y \)[/tex]-value [tex]\(19\)[/tex]. This means that for [tex]\( x = 3 \)[/tex], we have two different outputs ([tex]\(14\)[/tex] and [tex]\(19\)[/tex]), indicating that option C does not represent a function.
### Conclusion
Since neither Option B nor Option C satisfies the requirement that each [tex]\( x \)[/tex]-value must map to exactly one [tex]\( y \)[/tex]-value, neither option represents a function.
Therefore, the answer is:
```
None
```
