Question

Let $f\left(x\right)=x^2,x\in R.$ 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. $g\left(f\right(S\left)\right)\ne S$
• D. $g\left(f\right(S\left)\right)=g\left(S\right)$

Answer 1

The correct option is D $g\left(f\right(S\left)\right)=g\left(S\right)$

$f\left(S\right)=S^2\Rightarrow f\left(S\right)\in [0,16]$
$g\left(S\right)=\{x:x\in R,x^2\in S\}$
$\Rightarrow g\left(S\right)\in [-2,2]$
$g\left(f\right(S\left)\right)=\{x:f(x)\in f(S\left)\right\}=\{x:x^2\in [0,16\left]\right\}$
$\Rightarrow g\left(f\right(S\left)\right)\in [-4,4]$

$f\left(g\right(S\left)\right)=\left(g\right(S){)}^2\Rightarrow f(g\left(S\right)\in [0,4]$