Question

Let $f:R-\left\{n\right\}\to R$ 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

Let $x,y$ be two numbers in set $R-\left\{n\right\}$ such that $f\left(x\right)=f\left(y\right)$

$\therefore \frac{x-m}{x-n}=\frac{y-m}{y-n}$.

Simplifying the equation we get

$-my-nx=-mx-ny$

$\therefore y\left(n-m\right)=x\left(n-m\right)$

$\therefore y=x$

Hence the function is one-one.
Now for given $y\in R-\left\{1\right\}$, let $x=\frac{m-ny}{1-y}$.

Then for this $x$ we have $f\left(x\right)=y$. But there is no pre-image for $y=1$ Hence the function is into $R$.