A) f is one-one onto
B) f is one-one into
C) f is many-one onto
D) f is many-one into
Correct Answer: B
Solution :
[b] 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 \[f(x)=\alpha \] \[\Rightarrow a=\frac{x-m}{x-n}\Rightarrow (x-n)\alpha =x-m\] \[\Rightarrow x=\frac{n\alpha -m}{\alpha -1}.\] for \[\alpha =1,x\notin R\] so, f is not onto.
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