Concept of Functions

Question

!!f(x)+f[1/(1-x)]= (2(1-2x))/(x(1-x))!!

(a) !!1!!

(b) !!2!!

(c) !!3!!

(d) None

Solution

!!f(x)+f[1/(1-x)]= (2(1-2x))/(x(1-x))!!

Put !!x=2!!

!!f(2)+ f[1/(1-2)]= f(2)+f(-1)= (2(1-4))/(2(1-2))=3!!

!!=> f(2)+f(-1)= 3.....(1)!!

Put !!x=-1!!

!!f(-1)+f(1/2)= (2(1+2))/(-1(1+1))!!

!!=> f(-1)+f(1/2)= -3......(2)!!

Substracting equation (1) from (2), we get

!!f(2)-f(1/2)= 6....(3)!!

Put !!x= 1/2!!

!!f(1/2)+ f(2)= 0....(4)!!

Adding equation (3) and (4)

!!2f(2)=6!!

!!=> f(2)=3!!

Therefore, option (c) is correct.