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Simplifying f(x + y) = f(x) + f(y) + xy + 1 (x * f + y * f) = f(x) + f(y) + xy + 1 (fx + fy) = f(x) + f(y) + xy + 1 Multiply f * x fx + fy = fx + f(y) + xy + 1 Multiply f * y fx + fy = fx + fy + xy + 1 Reorder the terms: fx + fy = 1 + fx + fy + xy Add '-1fx' to each side of the equation. fx + -1fx + fy = 1 + fx + fy + -1fx + xy Combine like terms: fx + -1fx = 0 0 + fy = 1 + fx + fy + -1fx + xy fy = 1 + fx + fy + -1fx + xy Reorder the terms: fy = 1 + fx + -1fx + fy + xy Combine like terms: fx + -1fx = 0 fy = 1 + 0 + fy + xy fy = 1 + fy + xy Add '-1fy' to each side of the equation. fy + -1fy = 1 + fy + -1fy + xy Combine like terms: fy + -1fy = 0 0 = 1 + fy + -1fy + xy Combine like terms: fy + -1fy = 0 0 = 1 + 0 + xy 0 = 1 + xy Solving 0 = 1 + xy Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1xy' to each side of the equation. 0 + -1xy = 1 + xy + -1xy Remove the zero: -1xy = 1 + xy + -1xy Combine like terms: xy + -1xy = 0 -1xy = 1 + 0 -1xy = 1 Divide each side by '-1y'. x = -1y-1 Simplifying x = -1y-1
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