Question

If the function are defined as $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=\sqrt{1-x},$ then what is the common domain of the following functions: $f+g,f-g,\frac{f}{g},\frac{g}{f},g-f$ where $(f\pm g)\left(x\right)=f\left(x\right)\pm g\left(x\right),\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

• A. $0<x\le 1$
• B. $0\le x<1$
• C. $0\le x\le 1$
• D. $0<x<1$

Answer 1

The correct option is D $0<x<1$

$f+g=\sqrt{x}+\sqrt{1-x}$
$\Rightarrow x\ge 0$ & $1-x\ge 0\Rightarrow x\in [0,1]$
$f-g=\sqrt{x}-\sqrt{1-x}$
$\Rightarrow x\ge 0$ & $1-x\ge 0\Rightarrow x\in [0,1]$
$\frac{f}{g}=\frac{\sqrt{x}}{\sqrt{1-x}}$
$\Rightarrow x\ge 0$ & $1-x>0\Rightarrow x\in [0,1)$
$\frac{g}{f}=\frac{\sqrt{1-x}}{\sqrt{x}}$
$\Rightarrow 1-x\ge 0$ & $x>0\Rightarrow x\in (0,1]$
$g-f=\sqrt{1-x}-\sqrt{x}$
$\Rightarrow 1-x\ge 0$ & $x\ge 0\Rightarrow x\in [0,1]$
$\Rightarrow x\in (0,1)$