For x ∈ R - \0, 1\, let f_1(x) = 1/x, f_2(x) = 1 - x and… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 9 Jan · Shift I

For

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

, let

f_1(x) = \dfrac{1}{x}

f_2(x) = 1 - x

and

f_3(x) = \dfrac{1}{1-x}

be three given functions. If a function,

J(x)

satisfies

(f_2 \circ J \circ f_1)(x) = f_3(x)

, then

J(x)

is equal to

A.
$f_2(x)$
B.
$f_3(x)$
C.
$f_1(x)$
D.
$\dfrac{1}{x} f_3(x)$

Show answer and solution

Correct option: B

Solution

We need f₂(J(f₁(x))) = f₃(x).

Apply f₂⁻¹ = f₂ (since f₂ is its own inverse: f₂(f₂(x)) = x) on both sides:

J(f₁(x)) = f₂(f₃(x)) = 1 - 1/(1-x) = -x/(1-x) = x/(x-1)

Now substitute f₁(x) = 1/x, so let t = 1/x, meaning x = 1/t:

J(t) = (1/t)/(1/t - 1) = (1/t)/((1-t)/t) = 1/(1-t) = f₃(t)

So J(x) = f₃(x).

[JEE Main 2019 · 9 Jan · Shift I]

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