Concept:For f(f(x))=x to hold for all x, substitute f(x) into itself and simplify until the expression equals x.Explanation:Given f(x)=x+1αx, substitute f(x) into itself:f(f(x))=x+1αx+1α(x+1αx)=αx+x+1α2x.We need αx+x+1α2x=x.Cross-multiply to compare coefficients:α2x=x(αx+x+1).Expanding:α2x=(α+1)x2+x.For this to be true for all x, the coefficient of x2 must be zero:α+1=0⇒α=−1.Check: with α=−1, the denominator becomes (−1)x+x+1=1, so:f(f(x))=1x=x.Thus, the required value is α=−1.Answer:α=−1, i.e. option D.