Question

Let $f:R\to R$ be defined by $f\left(x\right)=2x+\left|x\right|$ Then prove that $f\left(2x\right)+f(-x)-f\left(x\right)=2\left|x\right|$.

Answer 1

$f\left(x\right)=2x+\left|x\right|$

$f\left(2x\right)=4x+\left|2x\right|=4x+2\left|x\right|$

$f(-x)=-2x+|-x|=-2x+\left|x\right|$

Now, $f\left(2x\right)+f(-x)-f\left(x\right)=4x+2\left|x\right|-2x+\left|x\right|-2x-\left|x\right|$

$=2\left|x\right|$