If the solve the problem

Question:

Le $f(\mathrm{x})=\mathrm{x}^{2}, \mathrm{x} \in \mathrm{R}$. For any $\mathrm{A} \subseteq \mathrm{R}$, define $\mathrm{g}(\mathrm{A})=\{\mathrm{x} \in \mathrm{R}, f(\mathrm{x}) \in \mathrm{A}\}$. If $\mathrm{S}=[0,4]$, then which one of the following statements is not true ?

  1. $f(\mathrm{~g}(\mathrm{~S})) \neq f(\mathrm{~S})$

  2. $f(g(S))=S$

  3. $g(f(S))=g(S)$

  4. $\mathrm{g}(f(\mathrm{~S})) \neq \mathrm{S}$

Correct Option: , 3

Solution:

$\mathrm{g}(\mathrm{S})=[-2,2]$

So, $\mathrm{f}(\mathrm{g}(\mathrm{S}))=[0,4]=\mathrm{S}$

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

Also, $g(f(S))=[-4,4] \neq g(S)$

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

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