If-f-x-y-f-x-f-y-for-real-x-y-and-f-0-0-Let-F-x-f-x-1-f-x-2-then-F-x-is-a-even-b-odd-c-neither-even-nor-odd-

Question Number 44350 by Necxx last updated on 27/Sep/18

I f f (x + y) = f (x) . f (y) f o r r e a l x, y

a n d f (0) \neq 0 . L e t F (x) = \frac{f (x)}{1 + {(f (x))}^{2}}

t h e n F (x) i s

a) e v e n b) o d d c) n e i t h e r e v e n n o r o d d

Commented by maxmathsup by imad last updated on 27/Sep/18

\Rightarrow f (x) . f (- x) = f (0) \Rightarrow f {(0)}^{2} = f (0) \Rightarrow f (0) = 1 \Rightarrow F (0) = \frac{f (0)}{1 + {(f (0))}^{2}} = \frac{1}{2}

d u e t o F (0) \neq 0 F c a n t b e o d d l e t s e e i f F i s e v e n

F (- x) = \frac{f (- x)}{1 + {(f (- x))}^{2}} = \frac{\frac{1}{f (x)}}{1 + {(\frac{1}{f (x)})}^{2}} = \frac{1}{f (x) {\frac{f^{2} (x) + 1}{f^{2} (x)}}} = \frac{f (x)}{1 + f^{2} (x)} = F (x)

s o F i s e v e n .

Commented by Necxx last updated on 27/Sep/18

t h a n k y o u s i r

Commented by Necxx last updated on 27/Sep/18

{t h a t}^{'} s i t

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