Question

If $f:R-\{-1,1\}\to R$ is defined by $f\left(x\right)=\frac{x}{x^2-1}$, verify whether $f$ is one-to-one or not.

Answer 1

Let $x$ and $y$ be two arbitary elements such that $f\left(x\right)=f\left(y\right)$

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

or ,$x{y}^2-x=x^2-y$

or , $xy(x-y)+(x-y)=0$

or, $(x-y)(1+xy)=0$

Either $x-y=0\Rightarrow x=y$

Again , $1+xy=0\Rightarrow 1+y^2=0\Rightarrow y^2=-1\Rightarrow y=\sqrt{-1}=i$

$\therefore F$ is not one – to-one mapping in the fives domains

As value of $Y$ and $X$ are not real

_______ proved