A real valued function f(x)  satisfies the functional equation f(x−y)=f(x)f(y)−f(a−x)f(a+y)  for some given constant a  and f(0)=1 , then f(2a−x)  is equal to :

f(x−y)=f(x)f(y)−f(a−x)f(a+y) x=y=0 ⇒ f(0)=f(0)2−(f(a))2 ⇒f(a)=0  as f(0)=1 f(2a−x)=f(a−(x−a))                     =f(a)f(x−a)−f(a−a)f(a+x−a)                     =−f(x) ∴   f(2a−x)=−f(x) .