Question

Let A=R-{2} and B=R-{1}, if f:A$\to$B is a mapping defined $f\left(x\right)=\frac{x-1}{x-2},$ show that f is one- one.

Answer 1

We have,

$A=R-\left\{2\right\}$ and $B=R-\left\{1\right\}$.If $A\to B$ is mapping defined.

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

Let $x,y\in A$ such that

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

$Now,$

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

$\Rightarrow (x-2)(y-3)=(y-2)(x-3)$

$\Rightarrow xy-3x-2y+6=xy-3y-2x+6$

$\Rightarrow -3x-2y=-3y-2x$

$\Rightarrow 3x-2x=3y-2y$

$\Rightarrow x=y$

Hence, f is one one