Let f:R R be a function defined by f(x) = x - m x - n , where m n… - Vidyadip

MCQ

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

A
$f$ is one-one onto
✓
$f$ is one-one into
C
$f$ is many one onto
D
$f$ is many one into

✓

Answer

Correct option: B.

$f$ is one-one into

b
(b) For any $x,\,y \in R,$ we have

$f(x) = f(y) \Rightarrow \frac{{x - m}}{{x - n}} = \frac{{y - m}}{{y - n}} \Rightarrow x = y$

$\therefore$ $f$ is one-one.

Let $\alpha$ $\in$ $R$ such that $f(x) = \alpha \Rightarrow \frac{{x - m}}{{x - n}} = \alpha $

==> $x = \frac{{m - n\alpha }}{{1 - \alpha }}$

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

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