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: A
Explanation
\(x * y = x + y - xy\) Let \(x^{-1}\) be the inverse of x, so that \(x * x^{-1} = x + x^{-1} - x(x^{-1}) = 0\) \(x + x^{-1} - x(x^{-1}) = 0 \implies x(x^{-1}) - x^{-1} = x\) \(x^{-1}(x - 1) = x \implies x^{-1} = \frac{x}{x - 1}\) = \(\frac{x}{-(1 - x)} = \frac{-x}{1 - x}, x \neq 1\)
