Question

If $f\left(x\right)=\left(\frac{x-1}{x+1}\right)$, then which of the following statements is/are correct?

• A. $f\left(\frac{1}{x}\right)=f\left(x\right)$
• B. $f\left(\frac{1}{x}\right)=-f\left(x\right)$
• C. $f(-\frac{1}{x})=\frac{1}{f\left(x\right)}$
• D. $f(-\frac{1}{x})=-\frac{1}{f\left(x\right)}$

Answer 1

Answer: (B), (D)

The correct options are
B $f\left(\frac{1}{x}\right)=-f\left(x\right)$
D $f(-\frac{1}{x})=-\frac{1}{f\left(x\right)}$

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

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

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

Hence, options B and D are correct.