Question

If $2f\left(x-1\right)-f\left(\frac{1-x}{x}\right)=x$ , then $f\left(x\right)$
=___________

• A. $\frac{1}{3}\left[2\left(1+x\right)+\frac{1}{1+x}\right]$
• B. $2\left(x-1\right)-\frac{1-x}{x}$
• C. $x^2+\frac{1}{x^2}+3$
• D. None of these

Answer 1

The correct option is A $\frac{1}{3}\left[2\left(1+x\right)+\frac{1}{1+x}\right]$

$2f(x-1)-f\left(\frac{1-x}{x}\right)=x$

put $x=1/x$

$-f(x-1)+2f\left(\frac{1-x}{x}\right)=1/x$

Multiply first equation by 2 and add first and second equation. We get,

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

put $x=x+1$

$3f\left(x\right)=2(x+1)+\frac{1}{(x+1)}$

$f\left(x\right)=\frac{2(x+1)+\frac{1}{(x+1)}}{3}$