f-x-f-1-x-x-2-gt-f-x-

Question Number 191151 by TUN last updated on 19/Apr/23

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

=> f (x) = ¿

Answered by Rasheed.Sindhi last updated on 19/Apr/23

f (x) + f (1 - x) = x^{2} \dots \dots . (i)

R e p l a c i n g x b y 1 - x :

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

f (1 - x) + f (x) = {(1 - x)}^{2} \dots \dots (i i)

(i) & (i i) : x^{2} = {(1 - x)}^{2}

x^{2} = 1 - 2 x + x^{2}

1 - 2 x = 0 \Rightarrow x = \frac{1}{2}

x i s c o n s t a n t

f (\frac{1}{2}) + f (1 - \frac{1}{2}) = {(\frac{1}{2})}^{2} = \frac{1}{4}

2 f (\frac{1}{2}) = \frac{1}{4} \Rightarrow f (\frac{1}{2}) = \frac{1}{8}

Commented by mr W last updated on 19/Apr/23

b u t i n f (x) + f (1 - x) = x^{2} x i s n o t a

c o n s t a n t b u t a v a r i a b l e (\in R) .

s o i t h i n k t h e a n s w e r i s t h a t

f (x) s a t i s f y i n g f (x) + f (1 - x) = x^{2}

{d o e s n}^{'} t e x i s t .

Commented by Rasheed.Sindhi last updated on 19/Apr/23

Very right sir! Thanks!