Question

Let $f,g:R\to R$ be two functions defined as $f\left(x\right)=|x|+x$ and $g\left(x\right)=|x|-x,\forall x\in R$. Then $(f\circ g)\left(x\right)$ for $x<0$ is

• A. $0$
• B. $4x$
• C. $-4x$
• D. $2x$

Answer 1

The correct option is C $-4x$

$f\left(x\right)=\{\begin{array}{cc}2x,& \forall x\ge 0\\ 0,& \forall x<0\end{array}$
$g\left(x\right)=|x|-x$
$\Rightarrow g\left(x\right)=-2x,x<0$
For $x<0$,
$(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)=f(-2x)$
$=2(-2x)[\because -2x>0forx<0]$
$=-4x$