Question

A real valued function f (x) satisfies the functional equation f(x – y) = f(x) f(y) – f(a – x) f(a + y) where a is a given constant and f(0) = 1.

f(2a – x) is equal to 

(a) f(x) 

(b) –f(x) 

(c) f(–x) 

(d) f(a) + f(a – x).

Answer 1

(b) : Given f(x – y) = f(x) f(y) – f(a – x) f(a + y)(*)
let x = 0 = y
f(0) = ( f(0))² – ( f(a))²
1 = 1 – ( f(a))² => f(a) = 0
Therefore, f(2a – x) = f(a – (x – a))

By using (*)

= f(a) f(x – a) – f(a + x – a)f (0)
= 0 – f(x)(1) = – f(x)  (∵ f(a) = 0, f(0) = 1)