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

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) !!

$$\mathrm{If}\: \\ $$$$\bullet\:{f}\left(\mathrm{2}{x}\:+\:\mathrm{1}\right)\:+\:{g}\left(\mathrm{3}\:−\:{x}\right)\:=\:{x} \\ $$$$\bullet\:{f}\left(\frac{\mathrm{3}{x}\:+\:\mathrm{5}}{\:{x}\:+\:\mathrm{1}}\right)\:+\:\mathrm{2}{g}\left(\frac{\mathrm{2}{x}\:+\:\mathrm{1}}{{x}\:+\:\mathrm{1}}\right)\:=\:\frac{{x}}{{x}\:+\:\mathrm{1}} \\ $$$$\mathrm{for}\:\mathrm{every}\:{x}\:\in\:\mathbb{R},\:\:{x}\:\neq\:−\mathrm{1} \\ $$$$\mathrm{Find}\:{f}\left({x}\right)\:!! \\ $$

Answered by ajfour last updated on 05/Mar/17

(((3x−7))/2)   is it?!

$$\frac{\left(\mathrm{3}{x}−\mathrm{7}\right)}{\mathrm{2}}\:\:\:{is}\:{it}?! \\ $$

Commented by ajfour last updated on 05/Mar/17

starting with the second,   f(3+(2/(x+1)))+2g(2−(1/(x+1))) = (((x+1)−1)/(x+1))  if (1/(x+1))→ y (we say)  f(3+2y)+2g(2−y)= ((1/y−1)/(1/y)) =1−y  if now y→ y−1, then  f(2y+1) +2 g(3−y) =2−y ;  or f(2x+1) +2g(3−x)= 2−x  subtracting this eqn from  2f(2x+1) +2g(3−x) =2x (in question)  we get: f(2x+1) =3x−2   if x→x−(1/2) ,  f(x)= 3(x−(1/2))−2  f(x) = (3x−7)/2 .

$${starting}\:{with}\:{the}\:{second},\: \\ $$$${f}\left(\mathrm{3}+\frac{\mathrm{2}}{{x}+\mathrm{1}}\right)+\mathrm{2}{g}\left(\mathrm{2}−\frac{\mathrm{1}}{{x}+\mathrm{1}}\right)\:=\:\frac{\left({x}+\mathrm{1}\right)−\mathrm{1}}{{x}+\mathrm{1}} \\ $$$${if}\:\frac{\mathrm{1}}{{x}+\mathrm{1}}\rightarrow\:{y}\:\left({we}\:{say}\right) \\ $$$${f}\left(\mathrm{3}+\mathrm{2}{y}\right)+\mathrm{2}{g}\left(\mathrm{2}−{y}\right)=\:\frac{\mathrm{1}/{y}−\mathrm{1}}{\mathrm{1}/{y}}\:=\mathrm{1}−{y} \\ $$$${if}\:{now}\:{y}\rightarrow\:{y}−\mathrm{1},\:{then} \\ $$$${f}\left(\mathrm{2}{y}+\mathrm{1}\right)\:+\mathrm{2}\:{g}\left(\mathrm{3}−{y}\right)\:=\mathrm{2}−{y}\:; \\ $$$${or}\:{f}\left(\mathrm{2}{x}+\mathrm{1}\right)\:+\mathrm{2}{g}\left(\mathrm{3}−{x}\right)=\:\mathrm{2}−{x} \\ $$$${subtracting}\:{this}\:{eqn}\:{from} \\ $$$$\mathrm{2}{f}\left(\mathrm{2}{x}+\mathrm{1}\right)\:+\mathrm{2}{g}\left(\mathrm{3}−{x}\right)\:=\mathrm{2}{x}\:\left({in}\:{question}\right) \\ $$$${we}\:{get}:\:{f}\left(\mathrm{2}{x}+\mathrm{1}\right)\:=\mathrm{3}{x}−\mathrm{2} \\ $$$$\:{if}\:{x}\rightarrow{x}−\frac{\mathrm{1}}{\mathrm{2}}\:,\:\:{f}\left({x}\right)=\:\mathrm{3}\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{2} \\ $$$${f}\left({x}\right)\:=\:\left(\mathrm{3}{x}−\mathrm{7}\right)/\mathrm{2}\:. \\ $$

Commented by Joel576 last updated on 08/Mar/17

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

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