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