Question

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

f is one-one into

Answer 2

Let $f : R \to R$ be a function defined by $f(x) = \frac{x -m}{x-n}$ For any $(x, y) \in R$ Let $f (x) = f (y)$ $\Rightarrow \frac{x-m}{x-n} = \frac{y-m}{y-n} \Rightarrow x =y$ $\therefore f$ 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.