QJEEM19Ma786f: JEE Main 2019 Mathematics - Class 11 JEEM19Ma786f Solution: Let f(x) = x 2 , x ∈ R. For any A ⊆ R, define g (A) = { x ∈ R : …

f(x) = x²    x $$\in$$ R

g(A) = {x $$\in$$ R : f(x) $$\in$$ A} S $$\equiv$$ [0, 4]

g(S) = {x $$\in$$ R : f(x) $$\in$$ S}

= {x $$\in$$ R : 0 $$\le$$ x² $$\le$$ 4}

= {x $$\in$$ R : –2 $$\le$$ x $$\le$$ 2}

$$\therefore$$ g(S) $$\ne$$ S

$$\therefore$$ f(g(S)) $$\ne$$ f(S)

g(f(S)) = {x $$\in$$ R : f(x) $$\in$$ f(S)}

= {x $$\in$$ R : x² $$\in$$ S²}

= {x $$\in$$ R : 0 $$\le$$ x² $$\le$$ 16}

= {x $$\in$$ R : –4 $$\le$$ x $$\le$$ 4}

$$\therefore$$ g(f(S)) $$\ne$$ g(S)

$$\therefore$$ g(f(S)) = g(S) is incorrect