To find the inverse of a relation, we simply swap each ordered pair, i.e., change the positions of the elements in each pair. Given the relation $R = \{ (-2, -7), (-1, -3), (0, 5), (1, 2) \}$, the inverse will have the form $R^{-1} = \{ (b, a) \}$, where each pair is reversed.

Inverse of $R$:

• The first pair is $(-2, -7)$, so its inverse will be $(-7, -2)$.
• The second pair is $(-1, -3)$, so its inverse will be $(-3, -1)$.
• The third pair is $(0, 5)$, so its inverse will be $(5, 0)$.
• The fourth pair is $(1, 2)$, so its inverse will be $(2, 1)$.

Therefore, the inverse relation $R^{-1}$ is: $R^{-1} = \{ (-7, -2), (-3, -1), (5, 0), (2, 1) \}$

Checking whether the inverse is a one-to-one function:

A relation is a one-to-one function (injective) if every element of the range (the second elements in the pairs) corresponds to exactly one element in the domain (the first elements in the pairs).

• In $R^{-1} = \{ (-7, -2), (-3, -1), (5, 0), (2, 1) \}$, all the second elements are distinct: $-2, -1, 0, 1$.
• Since all the second elements are unique, the inverse relation is a one-to-one function.

Summary:

• The inverse of $R$ is $R^{-1} = \{ (-7, -2), (-3, -1), (5, 0), (2, 1) \}$.
• Yes, the inverse is a one-to-one function because there are no repeated values in the second elements of the pairs.

Would you like further details on this process? Let me know if you have any questions!

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Here are some related questions to help explore this topic further:

1. How do you identify if a given relation is a function before finding its inverse?
2. What is the difference between a one-to-one function and a general function?
3. How do you check if a function is onto (surjective)?
4. How can you visualize the inverse of a function on a graph?
5. What happens if a relation contains repeated values in the second element of the pairs?

Tip: When working with inverse relations, always remember that the inverse of a function will only be a function if the original relation was one-to-one.