Answer :
To determine which option represents a function, we first need to understand the definition of a function in mathematical terms. A function is a relation where each input maps to exactly one output. In other words, for every [tex]\(x\)[/tex] value, there should be only one corresponding [tex]\(y\)[/tex] value.
Let's analyze each option provided:
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & -5 & 10 & 5 & -10 \\ \hline y & 13 & -7 & 23 & 17 & -17 \\ \hline \end{array} \][/tex]
In this table:
- The [tex]\(x\)[/tex] value 5 is paired with both 13 and 17.
Since the [tex]\(x\)[/tex] value 5 corresponds to two different [tex]\(y\)[/tex] values (13 and 17), Option A does not represent a function.
### Option B:
There are no details provided for Option B, making it impossible to analyze. Without data, we cannot determine if it represents a function.
### Option C:
Again, there are no details provided for Option C. Without data to inspect, we cannot decide if it represents a function.
### Option D:
[tex]\[ \{(-7, -9), (-4, -9), (5, 15), (7, 19)\} \][/tex]
In this set of ordered pairs, each [tex]\(x\)[/tex] value is paired with exactly one [tex]\(y\)[/tex] value:
- [tex]\(-7\)[/tex] maps to [tex]\(-9\)[/tex]
- [tex]\(-4\)[/tex] maps to [tex]\(-9\)[/tex]
- [tex]\(5\)[/tex] maps to [tex]\(15\)[/tex]
- [tex]\(7\)[/tex] maps to [tex]\(19\)[/tex]
Each [tex]\(x\)[/tex] value has a unique corresponding [tex]\(y\)[/tex] value, satisfying the definition of a function.
Therefore, Option D represents a function.
The correct answer is Option D.
Let's analyze each option provided:
### Option A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & -5 & 10 & 5 & -10 \\ \hline y & 13 & -7 & 23 & 17 & -17 \\ \hline \end{array} \][/tex]
In this table:
- The [tex]\(x\)[/tex] value 5 is paired with both 13 and 17.
Since the [tex]\(x\)[/tex] value 5 corresponds to two different [tex]\(y\)[/tex] values (13 and 17), Option A does not represent a function.
### Option B:
There are no details provided for Option B, making it impossible to analyze. Without data, we cannot determine if it represents a function.
### Option C:
Again, there are no details provided for Option C. Without data to inspect, we cannot decide if it represents a function.
### Option D:
[tex]\[ \{(-7, -9), (-4, -9), (5, 15), (7, 19)\} \][/tex]
In this set of ordered pairs, each [tex]\(x\)[/tex] value is paired with exactly one [tex]\(y\)[/tex] value:
- [tex]\(-7\)[/tex] maps to [tex]\(-9\)[/tex]
- [tex]\(-4\)[/tex] maps to [tex]\(-9\)[/tex]
- [tex]\(5\)[/tex] maps to [tex]\(15\)[/tex]
- [tex]\(7\)[/tex] maps to [tex]\(19\)[/tex]
Each [tex]\(x\)[/tex] value has a unique corresponding [tex]\(y\)[/tex] value, satisfying the definition of a function.
Therefore, Option D represents a function.
The correct answer is Option D.
