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A real valued function f(x) satisfies th...

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) =?

A

f(-x)

B

f(a)+f(a-x)

C

f(x)

D

`-f(x)`

Text Solution

Verified by Experts

The correct Answer is:
D
We have,
`f(x-y)=f(x)f(y)-f(a-y)f(a+y)` for all ` x, y in R ` …. (i)
Replacing x and y both by ) , we get
`f(0)={f(0)}^(2)-{f(a)}^(2)implies1=1-{f(a)}^(2)implies f(a)=0`
Now,
`f(2a-x)=f(a-(x-a))`
`implies f(2a-x)=f(a)f(x-a)-f(a-a)f(x)" "`[Replacing x by a and y by (x-a) in (i) ]
` f(2a-x) =-f(x)" " ` [ `:'` f(a)=0]
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