Answer :
Sure, let's determine which of the given sets of pairs and equations represent functions. A relation (set of pairs) is considered a function if each input value maps to exactly one output value, meaning no [tex]\( x \)[/tex]-values are repeated.
1. First Set:
[tex]\[ \{(-6, 17), (-2, 13), (2, 9), (6, 5), (10, 1)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-6, -2, 2, 6, 10\)[/tex].
- The [tex]\( x \)[/tex]-values are all unique.
Therefore, this set is a function.
2. Second Set:
[tex]\[ \{(-4, 2), (-2, 1), (-1, 3)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-4, -2, -1\)[/tex].
- The [tex]\( x \)[/tex]-values are all unique.
Therefore, this set is a function.
3. Third Set:
[tex]\[ \{(-3, 4), (0, 2), (0, 4), (3, 8), (4, 7)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-3, 0, 0, 3, 4\)[/tex].
- We see that the [tex]\( x \)[/tex]-value [tex]\( 0 \)[/tex] is repeated.
Therefore, this set is not a function.
4. Equation:
[tex]\[ y = -4x^2 + 45x + 9 \][/tex]
- This is a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- For any input [tex]\( x \)[/tex], there is a unique output [tex]\( y \)[/tex].
Therefore, the equation represents a function.
### Summary:
- First Set: Function
- Second Set: Function
- Third Set: Not a Function
- Equation: Function
So, the sets and equations that represent functions are:
- The set [tex]\(\{(-6, 17), (-2, 13), (2, 9), (6, 5), (10, 1)\}\)[/tex]
- The set [tex]\(\{(-4, 2), (-2, 1), (-1, 3)\}\)[/tex]
- The equation [tex]\( y = -4x^2 + 45x + 9 \)[/tex]
1. First Set:
[tex]\[ \{(-6, 17), (-2, 13), (2, 9), (6, 5), (10, 1)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-6, -2, 2, 6, 10\)[/tex].
- The [tex]\( x \)[/tex]-values are all unique.
Therefore, this set is a function.
2. Second Set:
[tex]\[ \{(-4, 2), (-2, 1), (-1, 3)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-4, -2, -1\)[/tex].
- The [tex]\( x \)[/tex]-values are all unique.
Therefore, this set is a function.
3. Third Set:
[tex]\[ \{(-3, 4), (0, 2), (0, 4), (3, 8), (4, 7)\} \][/tex]
- Here, we check the [tex]\( x \)[/tex]-values: [tex]\(-3, 0, 0, 3, 4\)[/tex].
- We see that the [tex]\( x \)[/tex]-value [tex]\( 0 \)[/tex] is repeated.
Therefore, this set is not a function.
4. Equation:
[tex]\[ y = -4x^2 + 45x + 9 \][/tex]
- This is a quadratic equation of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- For any input [tex]\( x \)[/tex], there is a unique output [tex]\( y \)[/tex].
Therefore, the equation represents a function.
### Summary:
- First Set: Function
- Second Set: Function
- Third Set: Not a Function
- Equation: Function
So, the sets and equations that represent functions are:
- The set [tex]\(\{(-6, 17), (-2, 13), (2, 9), (6, 5), (10, 1)\}\)[/tex]
- The set [tex]\(\{(-4, 2), (-2, 1), (-1, 3)\}\)[/tex]
- The equation [tex]\( y = -4x^2 + 45x + 9 \)[/tex]
