Paragraph: Read the following write up carefully and answer the following questions: Consider a function f: R R such that (f(x)+f(z))(f(y)+f(t))=f(x y-z t)+f(x t+y z) where x, y, z, t R Question: For the function, f: R R, f(x) can be

Step-by-step solution

aligned & Put y=z=t=0 & f(0)[f(x)+f(0)]=f(0) & Put x=0 & 2 f^2(0)=f(0) & f (0)=0, 1 2 & If f(0)= 1 2 f(x)+ 1 2 =1 & f(x)= 1 2 & If f(0)=0, z=t=0 & f(x) f(y)=f(x y) & Let, x=y=1 f^2(1)=f(1) & f(1)=0 or f(1)=1 & We have f(0)=0, f(1)=0, y=1 & f(x)=0 & Also, f(0)=0, f(1)=1, x=0, y=t=1 & (f(0)+f(z))(f(1)+f(1))=f(-z)+f(z) & 2 f(z)=f(-z)+f(z) & f ( z )= f (- z ) aligned