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Question Number 118513 by I want to learn more last updated on 18/Oct/20

If    f(x)  +  f(((x  −  1)/x))   =   1  +  x,       find   f(2).

$$\mathrm{If}\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:+\:\:\mathrm{f}\left(\frac{\mathrm{x}\:\:−\:\:\mathrm{1}}{\mathrm{x}}\right)\:\:\:=\:\:\:\mathrm{1}\:\:+\:\:\mathrm{x},\:\:\:\:\:\:\:\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right). \\ $$

Answered by Dwaipayan Shikari last updated on 18/Oct/20

f(2)+f(((2−1)/2))=1+2→(a)   (Takng x=2)  f((1/2))+f((((1/2)−1)/(1/2)))=1+(1/2)→(b)  (Taking x=(1/2))  f(2)−f(−1)=(3/2)→(a)−(b)  f(−1)+f(((−1−1)/(−1)))=1−1  →(f) (Taking  x=−1)    2f(2)=(3/2)  →(a)−(b)+(f)  f(2)=(3/4)

$${f}\left(\mathrm{2}\right)+{f}\left(\frac{\mathrm{2}−\mathrm{1}}{\mathrm{2}}\right)=\mathrm{1}+\mathrm{2}\rightarrow\left({a}\right)\:\:\:\left({Takng}\:{x}=\mathrm{2}\right) \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+{f}\left(\frac{\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{2}}}\right)=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\rightarrow\left({b}\right)\:\:\left({Taking}\:{x}=\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${f}\left(\mathrm{2}\right)−{f}\left(−\mathrm{1}\right)=\frac{\mathrm{3}}{\mathrm{2}}\rightarrow\left({a}\right)−\left({b}\right) \\ $$$${f}\left(−\mathrm{1}\right)+{f}\left(\frac{−\mathrm{1}−\mathrm{1}}{−\mathrm{1}}\right)=\mathrm{1}−\mathrm{1}\:\:\rightarrow\left({f}\right)\:\left({Taking}\:\:{x}=−\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{2}{f}\left(\mathrm{2}\right)=\frac{\mathrm{3}}{\mathrm{2}}\:\:\rightarrow\left({a}\right)−\left({b}\right)+\left({f}\right) \\ $$$${f}\left(\mathrm{2}\right)=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$ \\ $$

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir, but please i don′t understand the changement of nunbers.  like the second line and the like.

$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{but}\:\mathrm{please}\:\mathrm{i}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand}\:\mathrm{the}\:\mathrm{changement}\:\mathrm{of}\:\mathrm{nunbers}. \\ $$$$\mathrm{like}\:\mathrm{the}\:\mathrm{second}\:\mathrm{line}\:\mathrm{and}\:\mathrm{the}\:\mathrm{like}. \\ $$

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir, i understand now.

$$\mathrm{Thanks}\:\mathrm{sir},\:\mathrm{i}\:\mathrm{understand}\:\mathrm{now}. \\ $$

Answered by benjo_mathlover last updated on 18/Oct/20

 Given (1) f(x) + f (((x−1)/x)) = 1+x  replacing x by ((x−1)/x)  (2) f(((x−1)/x)) +f (((((x−1)/x)−1)/((x−1)/x))) = 1+((x−1)/x)  ⇒ f(((x−1)/x)) + f ((1/(1−x))) = ((2x−1)/x)  replacing ((x−1)/x) by x  (3) f ((1/(1−x))) + f(x) = 1+ (1/(1−x))  ⇒f ((1/(1−x))) + f(x) = ((2−x)/x)   adding (1),(2) and (3) ; gives   2 [ f(x) + f (((x−1)/x))+ f ((1/(1−x))) ]= 1+x +((2x−1)/x) + ((2−x)/x)  substitute equation (2)   2 [ f(x) + ((2x−1)/x) ] = ((x^2 +2x+1)/x)   f(x) = ((x^2 +2x+1)/(2x)) − ((2x−1)/x)   f(x) = ((x^2 −2x+3)/(2x)) ⇒f(2)= ((4−4+3)/4)  ⇒f(x) = (3/4)

$$\:{Given}\:\left(\mathrm{1}\right)\:{f}\left({x}\right)\:+\:{f}\:\left(\frac{{x}−\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}+{x} \\ $$$${replacing}\:{x}\:{by}\:\frac{{x}−\mathrm{1}}{{x}} \\ $$$$\left(\mathrm{2}\right)\:{f}\left(\frac{{x}−\mathrm{1}}{{x}}\right)\:+{f}\:\left(\frac{\frac{{x}−\mathrm{1}}{{x}}−\mathrm{1}}{\frac{{x}−\mathrm{1}}{{x}}}\right)\:=\:\mathrm{1}+\frac{{x}−\mathrm{1}}{{x}} \\ $$$$\Rightarrow\:{f}\left(\frac{{x}−\mathrm{1}}{{x}}\right)\:+\:{f}\:\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)\:=\:\frac{\mathrm{2}{x}−\mathrm{1}}{{x}} \\ $$$${replacing}\:\frac{{x}−\mathrm{1}}{{x}}\:{by}\:{x} \\ $$$$\left(\mathrm{3}\right)\:{f}\:\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)\:+\:{f}\left({x}\right)\:=\:\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1}−{x}} \\ $$$$\Rightarrow{f}\:\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)\:+\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}−{x}}{{x}}\: \\ $$$${adding}\:\left(\mathrm{1}\right),\left(\mathrm{2}\right)\:{and}\:\left(\mathrm{3}\right)\:;\:{gives}\: \\ $$$$\mathrm{2}\:\left[\:{f}\left({x}\right)\:+\:{f}\:\left(\frac{{x}−\mathrm{1}}{{x}}\right)+\:{f}\:\left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\right)\:\right]=\:\mathrm{1}+{x}\:+\frac{\mathrm{2}{x}−\mathrm{1}}{{x}}\:+\:\frac{\mathrm{2}−{x}}{{x}} \\ $$$${substitute}\:{equation}\:\left(\mathrm{2}\right) \\ $$$$\:\mathrm{2}\:\left[\:{f}\left({x}\right)\:+\:\frac{\mathrm{2}{x}−\mathrm{1}}{{x}}\:\right]\:=\:\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{{x}} \\ $$$$\:{f}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{\mathrm{2}{x}}\:−\:\frac{\mathrm{2}{x}−\mathrm{1}}{{x}} \\ $$$$\:{f}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{3}}{\mathrm{2}{x}}\:\Rightarrow{f}\left(\mathrm{2}\right)=\:\frac{\mathrm{4}−\mathrm{4}+\mathrm{3}}{\mathrm{4}} \\ $$$$\Rightarrow{f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir.

$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$

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