Functions Question 395
Question: 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
Options:
A) $ -f(x) $
B) $ f(x) $
C) $ f(a)+f(a-x) $
D) $ f(-x) $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ f(2a-x)=f(a-(x-a))=f(a)f(x-a)-f(0) $ $ f(x)=f(a)f(x-a)-f(x)=-f(x) $ $ [\because ,x=0,y=0,f(0)=f^{2}(0)-f^{2}(a) $
$ \Rightarrow f^{2}(a)=0 $
$ \Rightarrow f(a)=0]\Rightarrow f(2a-x)=-f(x) $
BETA
