The function f(x) defined on the real numbers has the property thatf(f(x)).(1+f(x)) = -f(x) for all x in the domain of f. If the number 3 is in domain and range of f. Compute the value of f(3).

f(f(x)).(1+f(x))=-f(x)Diff. wrt xf(f(x)).f`(x) + (1+f(x))f`(f(x))=-f`(x)Adding f`(x) on both sides and taking it common in 1st part of LHS[1+f(x)]f`(f(x)) + f`(x)[1+f(f(x))]=0Using properties of differentiation, LHS becomes[1+f(x)][1+f(f(x))]=0Therefore, 1+f(x)=0f(x)=-1f(3)=-1

Approved