Final answer:
The domain of the relation R = {(x, y): y = -x²} includes all real numbers, meaning any real number for x will produce a valid y-value. Therefore, the correct answer is (c) x ∈ R.
Step-by-step explanation:
The domain of a function consists of all the possible input values (x-values) that will produce a valid output (y-value) for that particular function. In the given relation R = {(x, y): y = -x²}, the equation represents a parabola that opens downwards. Since there is no restriction placed on the values of x in the equation, any real number for x will yield a real number for y. Therefore, the domain of R is all real numbers.
Looking at the options provided:
- (a) x ≥ 0, this suggests that only non-negative values of x are included. This is not correct because negative x-values also produce valid y-values.
- (b) x ≤ 0, this suggests that only non-positive values of x are included. This is not correct because positive x-values also produce valid y-values.
- (c) x ∈ R, this means that x is an element of real numbers, which is correct.
- (d) x ≠ 0, this would exclude zero from the domain, which is not the case as zero is a valid x-value in this relation.
Therefore, the correct answer is (c) x ∈ R, indicating that the domain of R includes all real numbers.