Question

If f(x)=(x-1)/(x+1) ,then show that (i) f(1/x)= -f(x) (ii) f(-1/x)= -1/f(x).

Answer 1

$given \: \: f(x) = \frac{x - 1}{x + 1} \\ (i) \: f( \frac{1}{x} ) = \frac{\frac{1}{x} - 1}{ \frac{1}{x}+ 1} = \frac{1 - x}{1 + x } \\ = \frac{ - (x - 1)}{x + 1} = - f(x) \\ \\ (ii) \: f( \frac{ - 1}{x} ) = \frac{\frac{ - 1}{x} - 1}{ \frac{ - 1}{x}+ 1} = \frac{ - 1 - x}{ - 1 + x } \\ = \frac{ -( 1 + x)}{ (x - 1) } = - 1 \times \frac{1}{ \frac{(x - 1)}{(x + 1)} } = - \frac{1}{f(x)}$