f-x-f-x-1-x-1-x-f-x-

Question Number 2205 by prakash jain last updated on 08/Nov/15

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

f (x) = ?

Commented by Yozzi last updated on 08/Nov/15

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

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

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

Answered by Rasheed Soomro last updated on 10/Nov/15

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

f (x) = 1 + x - f (\frac{x - 1}{x}) \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . (1)

f (\frac{x - 1}{x}) = 1 + \frac{x - 1}{x} - f (\frac{\frac{x - 1}{x} - 1}{\frac{x - 1}{x}}) [R e p l a c i n g x b y \frac{x - 1}{x} i n (1)]

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

f (\frac{- 1}{x - 1}) = 1 + \frac{- 1}{x - 1} - f (\frac{\frac{- 1}{x - 1} - 1}{\frac{- 1}{x - 1}}) [R e p l a c i n g x b y \frac{- 1}{x - 1} i n (1)]

= 1 - \frac{1}{x - 1} - f (x) \dots \dots \dots \dots \dots \dots \dots \dots \dots (3)

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R e p l a c i n g f (\frac{x - 1}{x}) b y r h s o f (2) a n d f (\frac{- 1}{x - 1}) b y r h s o f (3)

a n d f (x) b y r h s o f (1) i n (1)

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

f (x) = 1 + x - [1 + \frac{x - 1}{x} - f (\frac{- 1}{x - 1})]

f (x) = x - \frac{x - 1}{x} + f (\frac{- 1}{x - 1})]

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

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

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

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VERIFICATION :

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

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

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

f (x) + f (\frac{x - 1}{x}) = \frac{1}{2} (x - \overset{\times}{\frac{x - 1}{x}} - \overset{\times}{\frac{1}{x - 1}} + 1 + \overset{\times}{\frac{x - 1}{x}} + \overset{\times}{\frac{1}{x - 1}} + x + 1)

= \frac{1}{2} (2 x + 2)

= x + 1

Commented by Rasheed Soomro last updated on 11/Nov/15

f (x) i n v o l v e s f (\frac{x - 1}{x}), f (\frac{x - 1}{x}) i n v o l v e s f (\frac{- 1}{x - 1})

a n d f (\frac{- 1}{x - 1}) i n v o l v e s a g a i n f (x) . T h i s i s c y c l i c . A t l a s t

w e h a v e a n e q u a t i o n w h i c h i n v o l v e s o n l y f (x) w h i c h

g i v e s d e f i n i t i o n o f f (x) . I t h i n k t h i s c y c l i c t y p e

f u n c t i o n a l e q u a t i o n s h o w s a c a t e g a r y o f f u n c t i o n a l

e q u a t i o n s .

W e c a n a l s o t a l k o f o r d e r o f s u c h e q u a t i o n s .