	If R={(x,y) ; x, y ∈Z,x^2+3y^2≤8} is a relation on the set of integers Z, then the domain of R-1 is:
A. {0,1}		B. {-2,-1, 1, 2}
C. {-1,0,1}	D. {-2,-1,0,1,2}


The relation \( R = \{(x, y) \mid x, y \in \mathbb{Z}, x^2 + 3y^2 \leq 8\} \) is given. 

We need to find the domain of \( R^{-1} \), which is the inverse relation of \( R \). The domain of \( R^{-1} \) is the set of all values of \( y \) for which there exists at least one \( x \) such that \( (x, y) \) satisfies the condition \( x^2 + 3y^2 \leq 8 \).

### Let's examine the possible values of \( y \):

1. **\( y = 0 \):**
   - \( x^2 + 3(0)^2 \leq 8 \) simplifies to \( x^2 \leq 8 \).
   - \( x^2 \leq 8 \) implies \( x \) can be \( -2, -1, 0, 1, 2 \).

2. **\( y = 1 \):**
   - \( x^2 + 3(1)^2 \leq 8 \) simplifies to \( x^2 + 3 \leq 8 \), or \( x^2 \leq 5 \).
   - \( x^2 \leq 5 \) implies \( x \) can be \( -2, -1, 0, 1, 2 \).

3. **\( y = -1 \):**
   - \( x^2 + 3(-1)^2 \leq 8 \) is the same as the case for \( y = 1 \), so \( x \) can be \( -2, -1, 0, 1, 2 \).

4. **\( y = 2 \):**
   - \( x^2 + 3(2)^2 \leq 8 \) simplifies to \( x^2 + 12 \leq 8 \), which is not possible since \( x^2 \geq 0 \).

5. **\( y = -2 \):**
   - This case is similar to \( y = 2 \) and is also not possible.

Thus, the possible values for \( y \) are \( y = -1, 0, 1 \).

### Conclusion:
The domain of \( R^{-1} \) is \( \{-1, 0, 1\} \), which corresponds to option **C**.

**Answer:** C. \(\{-1, 0, 1\}\).

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Would you like further details or have any questions? Here are 8 related questions to consider:

1. What is the range of the original relation \( R \)?
2. How do you find the inverse of a relation in general?
3. How does the domain of \( R^{-1} \) relate to the range of \( R \)?
4. What are some properties of relations and their inverses?
5. How would the domain of \( R^{-1} \) change if the inequality were different?
6. What is the difference between a relation and a function?
7. How does one determine if a relation is symmetric?
8. Can \( R^{-1} \) be a function if \( R \) is not a function?

**Tip:** When dealing with inverse relations, the domain of the inverse is the range of the original relation.

Explore the domain of the inverse relation problem involving integer solutions. Understand how to determine the domain of the inverse relation based on given conditions. Suitable for high school students studying relations and functions.

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