Question

Find all real functions f from R → R satisfying the relation 

f(x² + yf(x)) = xf(x + y)

Answer 1

f(x² + yf(x)) = x.f(x + y) .........(1)

f(yf(0)) = 0 => f(0) = 0

for y = 0, x = – x

f(x²) = – xf(–x)

for y = 0 f(x²) = xf(x)

=> xf(–x) = – xf(x)