Mathematics - Functions Question with Solution | TestHub

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