Question

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

Answer 1

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