A real valued function f(x) satisfies the function equation f(x -… - Vidyadip

MCQ

A real valued function $f(x)$ satisfies the function 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

A
$f(a) + f(a - x)$
B
$f( - x)$
✓
$ - f(x)$
D
$f(x)$

✓

Answer

Correct option: C.

$ - f(x)$

c
(c) $f(a - (x - a)) = f(a)f(x - a) - f(0)f(x).....(i)$

Put $x = 0,y = 0$;

$f(0) = {(f(0))^2} - {[f(a)]^2} \Rightarrow f(a) = 0$

$[ \because f(0) =1 ].$ From $(i)$, $f(2a - x) = - f(x)$.

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