To determine which of the given relations is not a function, we need to recall the definition of a function. A relation is a function if every input (or first element of each ordered pair) maps to exactly one output (or second element of each ordered pair). Let's check each relation one by one.
1. Relation 1:
[tex]\[
\{(1,-1),(4,-2),(9,-3),(16,-4),(25,-5)\}
\][/tex]
In this relation, the first elements (inputs) are: 1, 4, 9, 16, and 25. Each input maps to a unique output:
- [tex]\( 1 \to -1 \)[/tex]
- [tex]\( 4 \to -2 \)[/tex]
- [tex]\( 9 \to -3 \)[/tex]
- [tex]\( 16 \to -4 \)[/tex]
- [tex]\( 25 \to -5 \)[/tex]
Since each input maps to one and only one output, Relation 1 is a function.
2. Relation 2:
[tex]\[
\{(1,1),(2,4),(3,9),(4,16),(5,25)\}
\][/tex]
In this relation, the first elements (inputs) are: 1, 2, 3, 4, and 5. Each input maps to a unique output:
- [tex]\( 1 \to 1 \)[/tex]
- [tex]\( 2 \to 4 \)[/tex]
- [tex]\( 3 \to 9 \)[/tex]
- [tex]\( 4 \to 16 \)[/tex]
- [tex]\( 5 \to 25 \)[/tex]
Since each input maps to one and only one output, Relation 2 is a function.
3. Relation 3:
[tex]\[
\{(1,-1),(2,-2),(0,0),(1,1),(2,2)\}
\][/tex]
In this relation, the first elements (inputs) are: 1, 2, 0, 1, and 2. Notice that:
- The input 1 maps to both -1 and 1.
- The input 2 maps to both -2 and 2.
Since an input (1 and 2) maps to more than one output, Relation 3 is not a function.
4. Relation 4:
[tex]\[
\{(-1,1),(-2,4),(-3,9),(-4,16),(-5,25)\}
\][/tex]
In this relation, the first elements (inputs) are: -1, -2, -3, -4, and -5. Each input maps to a unique output:
- [tex]\( -1 \to 1 \)[/tex]
- [tex]\( -2 \to 4 \)[/tex]
- [tex]\( -3 \to 9 \)[/tex]
- [tex]\( -4 \to 16 \)[/tex]
- [tex]\( -5 \to 25 \)[/tex]
Since each input maps to one and only one output, Relation 4 is a function.
Based on this detailed analysis, the relation that is not a function is:
[tex]\[
\{(1,-1),(2,-2),(0,0),(1,1),(2,2)\}
\][/tex]
Thus, the relation that is not a function is the third relation. The correct answer is 3.
