# Let $f(x)=x^{2},x\epsilon R.$ For any $A\subseteq R$ , define $g(A)={\left \{ x\epsilon R : f(x)\epsilon A}\right \}$. If $S=[0,4]$ , then whichone of the following statements is not true ? Option 1) $g(f(S)) \neq S$ Option 2) $f(g(S)) = S$ Option 3) $g(f(S)) = g(S)$ Option 4) $f(g(S)) \neq f(S)$

V Vakul

g(S) = [-2,2]

So, f(g(S)) = [0,4] = S

And f(S) = [0,16] => f(g(S))$\neq$ f(S)

Also,

g(f(S)) = [-4,4] $\neq$ g(S)

So, g(f(S))$\neq$ S

correct option is (3)

Option 1)

$g(f(S)) \neq S$

Option 2)

$f(g(S)) = S$

Option 3)

$g(f(S)) = g(S)$

Option 4)

$f(g(S)) \neq f(S)$

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