Question

Find the domain of definition of the following function:
$f\left(x\right)=\sqrt[4]{4x-|x|}+\log (x+2)$

Answer 1

$f\left(x\right)=\sqrt[4]{4x-|x|}+log(x+2)$

For $f\left(x\right)$ to be real and defined,

$x+2>0\Rightarrow x>-2$

$x-|x|$ must be$\ge$0.

if $x<0$

$x-|x|=2x<0$ which is not possible.

If $x\ge 0$

$x-|x|=0$

Therefore, domain of $f\left(x\right)$ is $x\in [0,\infty ).$