Question

If f(x) is continuous and increasing function such that domain of $g\left(x\right)=\sqrt{f\left(x\right)-x}$ be R and $h\left(x\right)=\frac{1}{1-x}$ , then the domain of $\varphi \left(x\right)=\sqrt{f\left(f\right(f\left(x\right)\left)\right)-h\left(h\right(h\left(x\right)\left)\right))}$ is

• A. $R$
• B. $\{0,1\}$
• C. $R-\{0,1\}$
• D. $R-(0,1)$

Answer 1

The correct option is C

$R-\{0,1\}$

$h\left(x\right)=\frac{1}{1-x},x\ne 1h\left(h\right(x\left)\right)=\frac{x-1}{x},x\ne 0,1\Rightarrow h\left(h\right(h\left(x\right)\left)\right)=x,x\ne 0,1$
Also $g\left(x\right)\ge 0\forall xϵR$
$\Rightarrow f\left(x\right)\ge x\Rightarrow f\left(f\right(x\left)\right)\ge f\left(x\right)\ge x$
($\because f\left(x\right)$ is an increasing function)
$\Rightarrow f\left(f\right(f\left(x\right)\left)\right)\ge f\left(f\right(x\left)\right)\ge f\left(x\right)\ge x\Rightarrow f\left(f\right(f\left(x\right)\left)\right)-x\ge 0\forall xϵR-\{0,1\}\Rightarrow \varphi \left(x\right)isdefinedforallx.ThusxϵR-\{0,1\}$