Prove Identities for \(f(x)\)

Solution

(i) Show that \(f\left(\frac1x\right)=-f(x)\)

Given: $$ f(x)=\frac{x-1}{x+1} $$

Replace \(x\) by \(\frac1x\): $$ f\left(\frac1x\right) = \frac{\frac1x-1}{\frac1x+1} $$

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

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

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

$$ =-f(x) $$

(ii) Show that \(f\left(-\frac1x\right)=-\frac1{f(x)}\)

Replace \(x\) by \(-\frac1x\): $$ f\left(-\frac1x\right) = \frac{-\frac1x-1}{-\frac1x+1} $$

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

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

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

Since $$ f(x)=\frac{x-1}{x+1}, $$

$$ \frac1{f(x)} = \frac{x+1}{x-1} $$

Therefore, $$ f\left(-\frac1x\right) = -\frac1{f(x)} $$

Hence proved.