Question

Let $f\left(x\right)=\{\begin{array}{cc}1,& \text{if x is a rational number}\\ 0,& \text{if x is an irrational number}\end{array}$

be a function from $R$ to $R$ where $R$ is the set of real numbers. Find

(i)$f\left(\frac{1}{3}\right)$

(ii)$f\left(\sqrt{7}\right)$

(iii)$f\left(fof\right)\left(1.4327\right)$

(iv)$f\left(fof\right)\left(\sqrt{3}\right)$

• A. $0,0,0,0$
• B. $1,1,1,1$
• C. $1,0,1,1$
• D. $0,1,0,1$

Answer 1

Answer: (C)

$1,0,1,1$

GIven that,

$f\left(x\right)=\{\begin{array}{cc}1,& \text{if x is a rational number}\\ 0,& \text{if x is an irrational number}\end{array}$

(i)$f\left(1/3\right)=f\left(\text{rational}\right)=1$

(ii)$f\left(\sqrt{7}\right)=f\left(\text{irrational}\right)=0$

(iii)$f\left(f0f\right)\left(1.4327\right)=fofof\left(\text{rational}\right)=fof\left(1\right)=f\left(1\right)=1$

(iv)$f\left(f0f\right)\left(\sqrt{3}\right)=fofof\left(\text{irrational}\right)=fof\left(0\right)=f\left(1\right)=1$