Let $f,g:\mathbb{R}\to \mathbb{R}$ be two functions defined as $f(x)=|x|+x$ and $g(x)=|x|-x$ for all $x\in \mathbb{R}$. The $(f\circ g)(x)$ for $x<0$ is:

Given $x<0$:Find $g(x)$:Since $x<0$, $|x|=-x$.
$$g(x)=|x|-x=-x-x=-2x$$
Since $x<0$, $-2x$ will be positive.Find $(f\circ g)(x)=f(g(x))=f(-2x)$:Since $-2x>0$, $|-2x|=-2x$.
$$f(-2x)=|-2x|+(-2x)=-2x-2x=-4x$$
Wait, let's re-evaluate: If $x=-1$, $g(-1)=|-1|-(-1)=1+1=2$.$f(2)=|2|+2=4$.Using $-4x$: $-4(-1)=4$. Correct.Calculation check from source: $|2x|+2x$ for $x<0$ where $2x$ is negative, so $|2x|=-2x$.Wait, the source calculation says: $||x|-x|+|x|-x$. For $x<0$, this is $|-x-x|+(-x)-x=|-2x|-2x$. Since $-2x$ is positive, $|-2x|=-2x$. So $-2x-2x=-4x$.Note: The source says $4x$ in the text but shows the logic for $-4x$. Let's stick to the math:For $x=-1$, $g(-1)=2$, $f(2)=4$.Option B ($4x$) gives $-4$. Option C ($-4x$) gives $4$.