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

Let f:R→Rf:R \rightarrow R be a function defined by f(x)=x−mx−n,f(x) = \frac{x - m}{x - n}, where m≠nm \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

f is one-one into

Explanation

Solution

For any x,y∈R,x,y \in R, we have

f(x)=f(y)⇒x−mx−n=y−my−n⇒x=yf(x) = f(y) \Rightarrow \frac{x - m}{x - n} = \frac{y - m}{y - n} \Rightarrow x = y ∴ f is one-one

Let α∈R\alpha \in R such that f(x)=α⇒x−mx−n=αf(x) = \alpha \Rightarrow \frac{x - m}{x - n} = \alpha ⇒ x=m−nα1−αx = \frac{m - n\alpha}{1 - \alpha}

Clearly x∉Rx \notin R for α=1\alpha = 1. So, f is not onto.