Let f: R arrow R be a function defined by f(x)=(x-m)/(x-n), where m ≠ n, then

Correct answer: B. f is one-one into Let f: R arrow R be a function defined by f(x)=(x-m)/(x-n) For any (x, y) ∈ R Let f(x)=f(y) (x-m)/(x-n)=(y-m)/(y-n) x=y f is one - one Let α ∈ R such that f(x)=α ⇒ α=(x-m)/(x-n) (x-n) α= x-m x α-n α=x-m x α- x=n α- m x(α-1)=n α-m x= n α-m α-1 · for α=1, x R So, f is not onto.

BITSAT 2014

Exam: BITSAT
Year: 2014
Subject: Mathematics
Chapter: Functions

BITSAT 2014 · Mathematics · Functions

Let f: R arrow R be a function defined by f(x)=(x-m)/(x-n), where m...

Let

\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}

be a function defined by

f(x)=\frac{x-m}{x-n},

where

m \neq n,

then

Options

A $\mathrm{f}$ is one-one onto
B $\mathrm{f}$ is one-one into
C $\mathrm{f}$ is many-one onto
Dfis many-one into

Correct answer

B. $\mathrm{f}$ is one-one into

Step-by-step solution

Let

f: R \rightarrow 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 ⇒ \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-

\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.

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