Functions Question 481
Question: Let $ f:R\to R $ be a function defined by $ f(x)=\frac{x-m}{x-n}, $ where $ m\ne n, $ then
Options:
A) f is one-one onto
B) f is one-one into
C) f is many-one onto
D) f is many-one into
Show Answer
Answer:
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.
BETA
