A real valued function f(x) satisfies the functional equationf(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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f(−x)

f(a)+fa−x

f(x)

−f(x)

answer is D.

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Detailed Solution

f(x−y)=f(x)f(y)−f(a−x)f(a+y)x=0,y=0⇒f(0−0)=f(0)f(0)−f(a−0)f(a+0)⇒f(0)=f02−fa2⇒1=12−fa2⇒fa2=0⇒f(a)=0Now f(2a−x)=f(a−(x−a))=f(a)f(x−a)−f(a−a)f(a+x−a)

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