Answer :
To determine which function has an inverse that is also a function, we need to check if each set of ordered pairs satisfies the property of being one-to-one (injective). A function is one-to-one if no two different elements in the domain map to the same element in the codomain. This means that each [tex]\( y \)[/tex]-value corresponds to exactly one [tex]\( x \)[/tex]-value.
Let's examine the given sets one by one:
1. Set 1: [tex]\( \{(-1, -2), (0, 4), (1, 3), (5, 14), (7, 4)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: [tex]\(-2, 4, 3, 14, 4\)[/tex].
- Notice that the value 4 repeats, corresponding to [tex]\( x = 0 \)[/tex] and [tex]\( x = 7 \)[/tex].
- Because the [tex]\( y \)[/tex]-value 4 maps to two different [tex]\( x \)[/tex]-values (0 and 7), this set does not have an inverse that is a function.
2. Set 2: [tex]\( \{(-1, 2), (0, 4), (1, 5), (5, 4), (7, 2)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 2, 4, 5, 4, 2.
- The values 2 and 4 both repeat, corresponding to multiple [tex]\( x \)[/tex]-values.
- Because the [tex]\( y \)[/tex]-values 2 and 4 each map to more than one [tex]\( x \)[/tex]-value, this set does not have an inverse that is a function.
3. Set 3: [tex]\( \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 3, 4, 14, 6, 2.
- All [tex]\( y \)[/tex]-values are unique.
- Since each [tex]\( y \)[/tex]-value corresponds to exactly one [tex]\( x \)[/tex]-value, this set has an inverse that is a function.
4. Set 4: [tex]\( \{(-1, 4), (0, 4), (1, 2), (5, 3), (7, 1)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 4, 4, 2, 3, 1.
- The value 4 repeats, corresponding to [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex].
- Because the [tex]\( y \)[/tex]-value 4 maps to two different [tex]\( x \)[/tex]-values (−1 and 0), this set does not have an inverse that is a function.
From our examination, we find that the set:
- [tex]\( \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \)[/tex] is the only set where each [tex]\( y \)[/tex]-value corresponds to exactly one [tex]\( x \)[/tex]-value, making its inverse a function.
Therefore, the answer is the third set:
[tex]\[ \{(-1,3),(0,4),(1,14),(5,6),(7,2)\} \][/tex]
Let's examine the given sets one by one:
1. Set 1: [tex]\( \{(-1, -2), (0, 4), (1, 3), (5, 14), (7, 4)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: [tex]\(-2, 4, 3, 14, 4\)[/tex].
- Notice that the value 4 repeats, corresponding to [tex]\( x = 0 \)[/tex] and [tex]\( x = 7 \)[/tex].
- Because the [tex]\( y \)[/tex]-value 4 maps to two different [tex]\( x \)[/tex]-values (0 and 7), this set does not have an inverse that is a function.
2. Set 2: [tex]\( \{(-1, 2), (0, 4), (1, 5), (5, 4), (7, 2)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 2, 4, 5, 4, 2.
- The values 2 and 4 both repeat, corresponding to multiple [tex]\( x \)[/tex]-values.
- Because the [tex]\( y \)[/tex]-values 2 and 4 each map to more than one [tex]\( x \)[/tex]-value, this set does not have an inverse that is a function.
3. Set 3: [tex]\( \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 3, 4, 14, 6, 2.
- All [tex]\( y \)[/tex]-values are unique.
- Since each [tex]\( y \)[/tex]-value corresponds to exactly one [tex]\( x \)[/tex]-value, this set has an inverse that is a function.
4. Set 4: [tex]\( \{(-1, 4), (0, 4), (1, 2), (5, 3), (7, 1)\} \)[/tex]
- The [tex]\( y \)[/tex]-values are: 4, 4, 2, 3, 1.
- The value 4 repeats, corresponding to [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex].
- Because the [tex]\( y \)[/tex]-value 4 maps to two different [tex]\( x \)[/tex]-values (−1 and 0), this set does not have an inverse that is a function.
From our examination, we find that the set:
- [tex]\( \{(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)\} \)[/tex] is the only set where each [tex]\( y \)[/tex]-value corresponds to exactly one [tex]\( x \)[/tex]-value, making its inverse a function.
Therefore, the answer is the third set:
[tex]\[ \{(-1,3),(0,4),(1,14),(5,6),(7,2)\} \][/tex]
