Question

If $f\left(x\right)=\frac{x+1}{x-1},$ then show that $f\left(x\right)+f\left(\begin{array}{c}1\\ x\end{array}\right)=0$.

Answer 1

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

$\Rightarrow f\left(x\right)+f\left(\frac{1}{x}\right)$

$\Rightarrow \frac{x+1}{x-1}+\frac{\frac{1}{x}+1}{\frac{1}{x}-1}$

$\Rightarrow \frac{x+1}{x-1}+\frac{x+1}{1-x}$

$\Rightarrow \frac{x+1}{x-1}-\frac{x+1}{x-1}$

$\Rightarrow 0$