For x ∈ R - \ 0, 1\ , let f _ 1 (x) = (1)/(x), f _ 2 (x) = 1 - x and f _ 3 (x) = (1)/(1 - x) be three given functions. If a function, J(x) satisfies ( f _ 2 oJo f _ 1 ) (x) = f _ 3 (x) then J(x) is equal to:

Correct answer: A. f _ 3 (x) Given f _ 2 (J( f _ 1 (x))) = f _ 3 (x) ⇒ f _ 2 (J((1)/(x))) = (1)/(1 - x) ⇒ 1 - J((1)/(x)) = (1)/(1 - x) ⇒ J((1)/(x)) = 1 - (1)/(1 - x) = - (x)/(1 - x) ⇒ J(x) = - ((1)/(x))/(1 - (1)/(x)) = - (1)/(x - 1) = (1)/(1 - x) ∴ J(x) = f _ 3 (x)

JEE Main 20199 JanuaryMorning Shift

Exam: JEE Main
Year: 2019
Date: 9 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Functions

JEE Main 2019 · 9 January · Morning Shift · Mathematics · Functions

For x ∈ R - \ 0, 1\ , let f _ 1 (x) = (1)/(x), f _ 2 (x) = 1 - x...

For

x \in R - \left\{0, 1\right\},

let

{f}_{1}\left(x\right) = \dfrac{1}{x}, {f}_{2}\left(x\right) = 1 - x

and

{f}_{3}\left(x\right) = \dfrac{1}{1 - x}

be three given functions. If a function,

J\left(x\right)

satisfies \left({f}_{2}oJo{f}_{1}\right)

\left(x\right)

= {f}_{3}\left(x\right)

then

J\left(x\right) is equal to:

Options

A ${f}_{3}\left(x\right)$
B $\dfrac{1}{x}{f}_{3}\left(x\right)$
C ${f}_{1}\left(x\right)$
D ${f}_{2}\left(x\right)$

Correct answer

A. ${f}_{3}\left(x\right)$

Step-by-step solution

Given ${f}_{2}\left(J\left({f}_{1}\left(x\right)\right)\right) = {f}_{3}\left(x\right)$

$⇒ {f}_{2}\left(J\left(\dfrac{1}{x}\right)\right) = \dfrac{1}{1 - x}$

$⇒ 1 - J\left(\dfrac{1}{x}\right) = \dfrac{1}{1 - x}$

$⇒ J\left(\dfrac{1}{x}\right) = 1 - \dfrac{1}{1 - x} = - \dfrac{x}{1 - x}$

$⇒ J\left(x\right) = - \dfrac{\dfrac{1}{x}}{1 - \dfrac{1}{x}}$

$= - \dfrac{1}{x - 1}$

$= \dfrac{1}{1 - x}$

$∴ J\left(x\right) = {f}_{3}\left(x\right)$

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