Given-that-f-0-0-for-all-x-y-R-2f-x-f-y-f-x-y-f-x-y-express-f-2x-in-term-of-f-x-

Question Number 106023 by ZiYangLee last updated on 02/Aug/20

Given that f (0) \neq 0 for all x, y \in R;

2 f (x) f (y) = f (x + y) + f (x - y),

express f (2 x) in term of f (x) .

Commented by mr W last updated on 02/Aug/20

y = 0 :

2 f (x) f (0) = f (x) + f (x)

\Rightarrow f (0) = 1

y = x :

2 f (x) f (x) = f (2 x) + f (0)

\Rightarrow f (2 x) = 2 {(f (x))}^{2} - 1

Commented by ZiYangLee last updated on 02/Aug/20

Wow Nice Thanks for your solutions

Answered by Aziztisffola last updated on 02/Aug/20

2 f (x) f (x) = f (2 x) + f (0)

f (2 x) = {2 f}^{2} (x) - f (0)