Question

which function has an inverse that is also a function?

\\(\\{(-4, 3), (-2, 7), (-1, 0), (4, -3), (11, -7)\\}\\)
\\(\\{(-4, 6), (-2, 2), (-1, 6), (4, 2), (11, 2)\\}\\)
\\(\\{(-4, 5), (-2, 9), (-1, 8), (4, 8), (11, 4)\\}\\)
\\(\\{(-4, 4), (-2, -1), (-1, 0), (4, 1), (11, 1)\\}\\)

Explanation:

Define the condition for an invertible function

Using the One-to-One Functions and Inverse Functions knowledge points

$$ \text{A function has an inverse that is also a function if and only if it is one-to-one (injective).} $$

$$ \forall (x_1, y_1), (x_2, y_2) \in f, \quad y_1 = y_2 \implies x_1 = x_2 $$

Analyze the first option

Using the One-to-One Functions knowledge point

$$ f_1 = \{(-4, 3), (-2, 7), (-1, 0), (4, -3), (11, -7)\} $$

$$ \text{Y-values: } \{3, 7, 0, -3, -7\} \quad (\text{all unique}) $$

Analyze the second option

Using the One-to-One Functions knowledge point

$$ f_2 = \{(-4, 6), (-2, 2), (-1, 6), (4, 2), (11, 2)\} $$

$$ \text{Repeated y-values: } 6 \text{ (from } -4 \text{ and } -1\text{), } 2 \text{ (from } -2\text{, } 4\text{, and } 11\text{)} $$

Analyze the third option

Using the One-to-One Functions knowledge point

$$ f_3 = \{(-4, 5), (-2, 9), (-1, 8), (4, 8), (11, 4)\} $$

$$ \text{Repeated y-value: } 8 \text{ (from } -1 \text{ and } 4\text{)} $$

Analyze the fourth option

Using the One-to-One Functions knowledge point

$$ f_4 = \{(-4, 4), (-2, -1), (-1, 0), (4, 1), (11, 1)\} $$

$$ \text{Repeated y-value: } 1 \text{ (from } 4 \text{ and } 11\text{)} $$

Answer:

• (A) \(\{(-4, 3), (-2, 7), (-1, 0), (4, -3), (11, -7)\}\) (Correct answer)
• (B) \(\{(-4, 6), (-2, 2), (-1, 6), (4, 2), (11, 2)\}\)
• (C) \(\{(-4, 5), (-2, 9), (-1, 8), (4, 8), (11, 4)\}\)
• (D) \(\{(-4, 4), (-2, -1), (-1, 0), (4, 1), (11, 1)\}\)