If-f-2x-1-g-3-x-x-f-3x-5-x-1-2g-2x-1-x-1-x-x-1-for-every-x-R-x-1-Find-f-x-

Question Number 10987 by Joel576 last updated on 05/Mar/17

If

∙ f (2 x + 1) + g (3 - x) = x

∙ f (\frac{3 x + 5}{x + 1}) + 2 g (\frac{2 x + 1}{x + 1}) = \frac{x}{x + 1}

for every x \in R, x \neq - 1

Find f (x)!!

Answered by ajfour last updated on 05/Mar/17

\frac{(3 x - 7)}{2} i s i t ?!

Commented by ajfour last updated on 05/Mar/17

s t a r t i n g w i t h t h e s e c o n d,

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

i f \frac{1}{x + 1} \to y (w e s a y)

f (3 + 2 y) + 2 g (2 - y) = \frac{1 / y - 1}{1 / y} = 1 - y

i f n o w y \to y - 1, t h e n

f (2 y + 1) + 2 g (3 - y) = 2 - y;

o r f (2 x + 1) + 2 g (3 - x) = 2 - x

s u b t r a c t i n g t h i s e q n f r o m

2 f (2 x + 1) + 2 g (3 - x) = 2 x (i n q u e s t i o n)

w e g e t : f (2 x + 1) = 3 x - 2

i f x \to x - \frac{1}{2}, f (x) = 3 (x - \frac{1}{2}) - 2

f (x) = (3 x - 7) / 2 .

Commented by Joel576 last updated on 08/Mar/17

t h a n k y o u v e r y m u c h

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