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JEE Main 201910 AprilMorning Shift
Exam
JEE Main
Year
2019
Date
10 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2019 · 10 April · Morning Shift · Mathematics · Functions

Let f(x) = x ^ 2 , x ∈ R . For any A ⊆ R, define g(A) = x ∈ R: f (...

Let f(x)=x2,x∈Rf\left(x\right) = {x}^{2}, x \in R . For any A⊆R,A ⊆ R, define g(A)=x∈R:f(x)∈Ag\left(A\right) = { x \in R: f ( x ) \in A } . If S=[0,4]S = [ 0, 4 ] , then which one of the following statements is not true?

Options

  1. Ag(f(S))≠Sg ( f ( S ) ) \ne S
  2. Bf(g(S))≠f(S)f ( g ( S ) ) \ne f ( S )
  3. Cf(g(S))=Sf\left(g\left(S\right)\right) = S
  4. Dg(f(S))=g(S)g\left(f\left(S\right)\right) = g\left(S\right)

Correct answer

D. g(f(S))=g(S)g\left(f\left(S\right)\right) = g\left(S\right)

Step-by-step solution

From the given information we can conclude
gg\left(S\right) = [−2,2]\left[- 2,2\right] , So, f(g(S))=[0,4]=Sf ( g ( S ) ) = \left[0, 4\right] = S
And f(S)=[0,16]⇒f(g(S))≠f(S)f ( S ) = \left[0, 16\right] ⇒ f\left(g\left(S\right)\right) \ne f\left(S\right)
Also, gg\left(f\left(S\right)\right) = \left[- 4, 4\right] \ne g\left(S\right)
So, gg\left(f\left(S\right)\right) \ne S
Hence, g(f(S))=g(S)g\left(f\left(S\right)\right) = g\left(S\right) is not true.
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