A binary operation * is defined on the set of real numbers, R, by (x * y = x + y – xy). If the identity element under the operation * is 0, find the inverse of (x in R).
-
A.
(frac{-x}{1 – x}, x neq 1) -
B.
(frac{1}{1 – x}, x neq 1) -
C.
(frac{-1}{1 – x}, x neq 1) -
D.
(frac{x}{1 – x}, x neq 1)
Correct Answer: Option A
Explanation
(x * y = x + y – xy)
Let (x^{-1}) be the inverse of x, so that
(x * x^{-1} = x + x^{-1} – x(x^{-1}) = 0)
(x + x^{-1} – x(x^{-1}) = 0 implies x(x^{-1}) – x^{-1} = x)
(x^{-1}(x – 1) = x implies x^{-1} = frac{x}{x – 1})
= (frac{x}{-(1 – x)} = frac{-x}{1 – x}, x neq 1)