If the functions are defined as
Question:

If the functions are defined as $f(x)=\sqrt{x}$ and $\mathrm{g}(\mathrm{x})=\sqrt{1-\mathrm{x}}$, then what is the common domain of the following functions:

$f+g, f-g, f / g, g / f, g-f$ where

$(f \pm g)(x)=f(\mathrm{x}) \pm \mathrm{g}(\mathrm{x}),(f / \mathrm{g})(\mathrm{x})=\frac{f(\mathrm{x})}{\mathrm{g}(\mathrm{x})}$

  1. (1) $0 \leq x \leq 1$

  2. (2) $0 \leq x<1$

  3. (3) $\$ 0$

  4. (4) $\$ 0$


Correct Option: , 3

Solution:

$f(x)+g(x)=\sqrt{x}+\sqrt{1-x}$, domain $[0,1]$

$f(x)-g(x)=\sqrt{x}-\sqrt{1-x}$, domain $[0,1]$

$\mathrm{g}(\mathrm{x})-f(\mathrm{x})=\sqrt{1-\mathrm{x}}-\sqrt{\mathrm{x}}$, domain $[0,1]$

$\frac{f(x)}{g(x)}=\frac{\sqrt{x}}{\sqrt{1-x}}$, domain $[0,1)$

$\frac{g(x)}{f(x)}=\frac{\sqrt{1-x}}{\sqrt{x}}$, domain $(0,1]$

So, common domain is $(0,1)$

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