Question

If $f\left(x\right)=\{\begin{array}{c}x,x\text{is rational}\\ 1-x,x\text{is irrational}\end{array}$,
then $f\left(f\right(x\left)\right)$ is

• A. $x\forall x\in R$
• B. $f\left(x\right)=\{\begin{array}{c}x,x\text{is irrational}\\ 1-x,x\text{is rational}\end{array}$
• C. $f\left(x\right)=\{\begin{array}{c}x,x\text{is rational}\\ 1-x,x\text{is irrational}\end{array}$
• D. None of these

Answer 1

The correct option is A $x\forall x\in R$

$f\left(f\right(x\left)\right)=\{\begin{array}{c}f\left(x\right),f\left(x\right)\text{is rational}\\ 1-f\left(x\right),f\left(x\right)\text{is irrational}\end{array}$
$\Rightarrow f\left(f\right(x\left)\right)=\{\begin{array}{c}x,x\text{is rational}\\ 1-(1-x)=x,x\text{is irrational}\end{array}$