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

Question Number 171955 by infinityaction last updated on 22/Jun/22

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

f (x) = ? ?

Answered by thfchristopher last updated on 22/Jun/22

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

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

= \frac{- {(1 - x)}^{2} + 2 - 3 x}{x (1 - x)}

= \frac{2 - 3 x}{x (1 - x)} - \frac{1 - x}{x}

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

= \frac{x + 1}{x} - \frac{1}{1 - x}

Comparison by terms,

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

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

Commented by infinityaction last updated on 22/Jun/22

i k n o w s o l u t i o n o f t h i s q u e s t i o n

b u t y o u r s o l u t i o n i s w r o n g

Commented by infinityaction last updated on 22/Jun/22

i n e e d a n y o t h e r w a y o f t h i s

q u e s t i o n

Commented by infinityaction last updated on 22/Jun/22

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

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

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

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

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

Commented by thfchristopher last updated on 22/Jun/22

Haha, I just tried on error . I certainly know

my answer should be gussed wrong .

Answered by mr W last updated on 22/Jun/22

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

r e p l a c e x i n (i) w i t h \frac{1}{1 - x} :

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

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

r e p l a c e x i n (i) w i t h \frac{x - 1}{x} :

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

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

(i) + (i i i) - (i i) :

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

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

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

Commented by infinityaction last updated on 22/Jun/22

t h a n k y o u s i r

Commented by peter frank last updated on 22/Jun/22

thank you

Commented by Tawa11 last updated on 25/Jun/22

Great sir

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