Question

Show that $f:[-1,1]\to R$, given by $f\left(x\right)=\frac{x}{(x+2)}$ is one-one. Find the inverse of the function $f:[-1,1]\to Rangef.$

Answer 1

Given $f:[-1,1]\to R$

$f\left(x\right)=\frac{x}{x+2}$

To show, one-one we have to show , if two different element of Domain gives same, then they are equal

Let $x,y\in [-1,1]$

Let $f\left(x\right)=f\left(y\right)$

$\Rightarrow \frac{x}{x+2}=\frac{y}{y+2}$

$\Rightarrow xy+2x=xy+2y$

$\Rightarrow 2x=2y$

$\Rightarrow x=y$

Hence they function is one-one

Now let $y=\frac{x}{x+2}\Rightarrow yx+2y=x$

$\Rightarrow x(y-1)=-2y$

$\Rightarrow x=\frac{2y}{1-y}$

$x=f^{-1}\left(\frac{x}{x+2}\right)=\frac{2y}{1-y}$