Question
A real valued function $$f\left( x \right)$$ satisfies the functional equation $$f\left( {x - y} \right) = f\left( x \right)f\left( y \right) - f\left( {a - x} \right)f\left( {a + y} \right)$$
where $$a$$ is a given constant and $$f\left( 0 \right) = 1,f\left( {2a - x} \right)$$ is equal to
A.
$$ - f\left( x \right)$$
B.
$$f\left( x \right)$$
C.
$$f\left( a \right) + f\left( {a - x} \right)$$
D.
$$f\left( { - x} \right)$$
Answer :
$$ - f\left( x \right)$$
Solution :
$$\eqalign{
& f\left( {2a - x} \right) = f\left( {a - \left( {x - a} \right)} \right) \cr
& = f\left( a \right)f\left( {x - a} \right) - f\left( 0 \right)f\left( x \right) = f\left( a \right)f\left( {x - a} \right) - f\left( x \right) \cr
& = - f\left( x \right)\,\left[ {\because x = 0,y = 0,f\left( 0 \right) = {f^2}\left( 0 \right) - {f^2}\left( a \right)} \right. \cr
& \left. { \Rightarrow {f^2}\left( a \right) = 0 \Rightarrow f\left( 0 \right) = 0} \right] \cr
& \Rightarrow f\left( {2a - x} \right) = - f\left( x \right) \cr} $$