Let $A=\{x \in R: x$ is not a positive integer $\}$ Define a function $f: A \rightarrow R$ as $f(x)=\frac{2 x}{x-1}$ then $f$ is

(A) injective but nor surjective

(C) surjective but not injective

(D) neither injective nor surjective

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Sets Relations and Functions

Let

A = {x \in R : x

is not a positive integer

}

Define a function

f : A \to R

f (x) = \frac{2 x}{x - 1}

then

f

Answer: (A)

Rewrite

f (x) = \frac{2 x}{x - 1} = 2 + \frac{2}{x - 1}

. Its derivative

f^{'} (x) = - \frac{2}{( x - 1 ) ^{2}}

is negative throughout the domain, so

f

is strictly decreasing and hence one-one (injective). For onto, solve

y = \frac{2 x}{x - 1}

for

x = \frac{y}{y - 2}

; the value

y = 2

yields no

x

, so

2

is never attained and the range misses part of

R

. Therefore

f

is injective but not surjective.

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option