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A real valued function satisfies the functional eq n f(x-y)=f(x)f(y)-f(a-x)f(a+y) where a is a constant and f(0)=1, f(2a-x) is equal to??? Explain.

A real valued function satisfies the functional eqn f(x-y)=f(x)f(y)-f(a-x)f(a+y) where a is a constant and f(0)=1, f(2a-x) is equal to??? Explain.

Grade:12th Pass

1 Answers

Swapnil Saxena
102 Points
9 years ago

The answer is -f(x)

Solution:If f(x-y)=f(x)f(y)-f(a-x)f(a+y)

            So f(a)=f(a-0)=f(a)f(0)-f(a-a)f(a+0)

            If f(0)=1 ,=f(a)-f(a)=0

            Then f(2a-x)=f(a+(a-x))=f(a-(x-a))=f(a)f(x-a)-f(a-a)f(a+x-a)=f(a)f(x-a)-f(0)f(x)=f(a)f(x-a)-f(x)

            If f(a)=0,the equtaion is reduced to -f(x)

 

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