Solve this following

Question:

If the functions are defined as $f(\mathrm{x})=\sqrt{\mathrm{x}}$ and

$g(x)=\sqrt{1-x}$, then what is the common domain of the following functions :

$f+\mathrm{g}, f-\mathrm{g}, f / \mathrm{g}, \mathrm{g} / f, \mathrm{~g}-f$ where $(f \pm \mathrm{g})(\mathrm{x})=$

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

 

  1. $0

  2. $0 \leq x<1$

  3. $0

  4. $0


Correct Option: , 3

Solution:

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

$f(\mathrm{x})-\mathrm{g}(\mathrm{x})=\sqrt{\mathrm{x}}-\sqrt{1-\mathrm{x}}$, 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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