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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 given consant and $f (0) = 1, f (2 a - x)$ is equal to

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- f (x)

f (- x)

f (x)

f (a) + f (a - x)

answer is D.

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Detailed Solution

f (x - y) = f (x) f (y) - f (a - x) f (a + y)

$f (a - 0) = f (a) f (0) - f (0) f (a)$

$f (a) = f (a) - f (a) (∴ f (0) = 1)$

$f (a) = 0$

$f (2 a - x) = f (a - (x - a))$

$= f (a) f (x - a) - f (0) f (x)$

$= 0 - f (x) = - f (x)$

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