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Q.

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

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a

$f (a) + f (a - x)$

b

$- f (x)$

c

$f (- x)$

d

$f (x)$

answer is D.

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Detailed Solution

$\begin{array}{l} f (x - y) = f (x) f (y) - f (a - x) . f (a + y) - - (1) \\ G i v e n f (0) = 1 \\ p u t x = y = 0 i n (1) \\ \Rightarrow f (0) = f (0) . f (0) - f (a) . f (a) \\ \Rightarrow 1 = 1 - {[f (a)]}^{2} \Rightarrow f (a) = 0 \end{array}$

$\begin{array}{l} N o w f (2 a - x) = f (a - (x - a)) \\ \Rightarrow f (a) . f (x - a) - f (a - a) . f (a + x - a) \\ \Rightarrow f (a) . f (x - a) - f (0) f (x) \\ \Rightarrow 0 - f (x) \\ \Rightarrow - f (x) \end{array}$

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