Question

If $f\left(x\right)=\frac{x-1}{x+1},$ then $f\left(\frac{1}{x}\right)+f\left(x\right)$ is equal to __________ .

Answer 1

$$\begin{gathered} \mathrm{If}f\left(x\right)=\frac{x-1}{x+1} \\ f\left(\frac{1}{x}\right)=\frac{{\displaystyle \frac{1}{x}}-1}{{\displaystyle \frac{1}{x}}+1} \\ =\frac{{\displaystyle \frac{\left(1-x\right)}{x}}}{{\displaystyle \frac{\left(1+x\right)}{x}}} \\ f\left(\frac{1}{x}\right)=\frac{1-x}{1+x} \\ \therefore f\left(\frac{1}{x}\right)+f\left(x\right)=\frac{1-x}{1+x}+\frac{x-1}{x+1} \\ =\frac{1-x+x-1}{x+1} \\ \therefore f\left(\frac{1}{x}\right)+f\left(x\right)=0 \end{gathered}$$