Question

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

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

Answer 1

The correct option is A $f$ is one one into function

Given $f\left(x\right)=\frac{x-m}{x-n}$ where $m\ne n,\forall x\in R-\left(n\right)$

Let $x_1,x_2\in R$

$\therefore f\left(x_1\right)=f\left(x_2\right)$

$\Rightarrow \frac{x_1-m}{x_1-n}=\frac{x_2-m}{x_2-n}\Rightarrow x_1=x_2$

$\therefore f$ is one-one. Let $\lambda \in R$

such that $f\left(x\right)=\lambda$

$\therefore \frac{x-m}{x-n}=\lambda$

$\therefore x=\frac{m-n\lambda }{1-\lambda }$, $x$ is not defined for $\lambda =1,$ also $x$ is not real.

$\therefore f\left(x\right)$ is not onto function.If a function is not onto it referred that it is into function.

Hence $f$ is one-one into function.
Ans: A