Question

if f(x)= x-1/x+1 then show that f(1/x)= -f(x)

Answer 1

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

Answer 2

Answer:  The proof is done below.

Step-by-step explanation:  We are given the following function :

$f(x)=\dfrac{x-1}{x+1}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We are to prove that :

$f\left(\dfrac{1}{x}\right)=-f(x).$

Replacing x by $\dfrac{1}{x}$ in equation (i), we have

$f\left(\dfrac{1}{x}\right)\\\\\\=\dfrac{\dfrac{1}{x}-1}{\dfrac{1}{x}+1}\\\\\\=\dfrac{\dfrac{1-x}{x}}{\dfrac{1+x}{x}}\\\\\\=\dfrac{1-x}{x}\times\dfrac{x}{1+x}\\\\\\=\dfrac{1-x}{1+x}\\\\=-\dfrac{x-1}{x+1}\\\\=-f(x).$

Thus, we get

$f\left(\dfrac{1}{x}\right)=-f(x).$

Hence proved.