Question

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

• A. $f\left(g\right(S\left)\right)\ne f\left(S\right)$
• B. $f\left(g\right(S\left)\right)=S$
• C. $g\left(f\right(S\left)\right)=f\left(S\right)$
• D. $g\left(f\right(S\left)\right)=\ne S$

Answer 1

Answer: (C)

$g\left(f\right(S\left)\right)=f\left(S\right)$

$g\left(S\right)=[-2,2]$
So, $f\left(g\right(S\left)\right)=[0,4]=S$
And $f\left(S\right)=[0,16]\Rightarrow f\left(g\right(S\left)\right)\ne f\left(S\right)$
Also, $g\left(f\right(S\left)\right)=[-4,4]\ne g\left(S\right)$
So, $g\left(f\right(S\left)\right)\ne S$