To determine which of the given relations are functions, we need to apply the definition of a function. A relation is a function if each input (or "x" value) has exactly one output (or "y" value).
Let's analyze each relation one by one:
### Relation 1:
[tex]\(\{(3,-2),(4,-2),(5,-2),(6,-2)\}\)[/tex]
- The pairs in this set are: [tex]\((3,-2), (4,-2), (5,-2), (6,-2)\)[/tex].
- Here, every x-value (3, 4, 5, 6) appears only once, each with exactly one corresponding y-value (-2).
Since each x-value is unique and maps to only one y-value, this relation is indeed a function.
### Relation 2:
[tex]\[
\begin{array}{cccccc}
x & 10 & 20 & 20 & -20 & -10 \\
y & -2 & -3 & -4 & -5 & -6 \\
\end{array}
\][/tex]
- There are pairs: (10, -2), (20, -3), (20, -4), (-20, -5), (-10, -6).
- Notice the x-value 20 appears twice with different y-values (-3 and -4).
Since the x-value 20 maps to two different y-values (-3 and -4), this relation is not a function.
### Relation 3:
[tex]\[ y = x^2 - 5 \][/tex]
This is an equation, and we need to determine whether it represents a function.
- For any input x, there is only one output y because squaring a number and subtracting 5 will always give a unique result.
Therefore, the equation [tex]\( y = x^2 - 5 \)[/tex] is a function.
### Conclusion
The relations that are functions are:
1. [tex]\(\{(3,-2),(4,-2),(5,-2),(6,-2)\}\)[/tex]
2. [tex]\( y = x^2 - 5 \)[/tex]
So, the correct answers are:
- [tex]\(\{(3,-2),(4,-2),(5,-2),(6,-2)\}\)[/tex]
- [tex]\( y = x^2 - 5 \)[/tex]
