Question

Let $f:R\to R$ 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

Answer: (B)

Let $f:R\to R$ be a function defined by
$f\left(x\right)=\frac{x-m}{x-n}$
For any $\left(x,y\right)\in R$
Let $f\left(x\right)=f\left(y\right)$
$\Rightarrow \frac{x-m}{x-n}=\frac{y-m}{y-n}\Rightarrow x=y$
$\therefore$ f is one - one
Let $\alpha \in R$ such that $f\left(x\right)=\alpha$
$\Rightarrow \alpha =\frac{x-m}{x-n}\Rightarrow \left(x-n\right)\alpha =x-m$
$\Rightarrow x\alpha -n\alpha =x-m\Rightarrow x\alpha -x=n\alpha -m$
$\Rightarrow x\left(\alpha -1\right)=n\alpha -m$
$\Rightarrow x=\frac{n\alpha -m}{\alpha -1}$. for $\alpha =1,x\notin R$
So, f is not onto.