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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(-x)

b

f(x)

c

–f(x)

d

f(a) + f(a – x)

answer is A.

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Detailed Solution

f(xy)=f(x)f(y)f(ax)f(a+y)x=0=yf(0)=[f(0)]2[f(a)]21=1[f(a)]2 [ f(0)=1]f(a)=0f(2ax)=f{a(xa)}=f(a)f(xa)f(aa)f(x)=0f(x)1=f(x)

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