Question

Let $f:R-\left\{x\right\}\to R$ be a function defined by $f\left(x\right)=\frac{x-m}{x-n}$, where $m\ne n$. Then

• A. $f$ is one-one onto
• B. $f$ is one-one into
• C. $f$ is many one onto
• D. $f$ is many one into

Answer 1

The correct option is B $f$ is one-one into

For any $x,y\in R$, we have
$f\left(x\right)=f\left(y\right)$
$\Rightarrow \frac{x-m}{x-n}=\frac{y-m}{y-n}$
$\Rightarrow x=y$
So, $f$ is one-one.
Let $\alpha ϵR$ such that $f\left(x\right)=\alpha$
$\Rightarrow \frac{x-m}{x-n}=\alpha$

$\Rightarrow x=\frac{m-n\alpha }{1-\alpha }$
Clearly, $x\notin R$ for $\alpha =1$.

So, $f$ is not onto.