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(D) Step $1$: Calculate $f(x)$ and $g(x)$ for specific values.For $x \ge 0$, $f(x) = 2x - x = x$ and $g(x) = 2x + x = 3x$.For $x < 0$, $f(x) = 2x - (-

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$(gof)(2) + (gof)(-2) = 0$

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(D) Step $1$: Calculate $f(x)$ and $g(x)$ for specific values.For $x \ge 0$, $f(x) = 2x - x = x$ and $g(x) = 2x + x = 3x$.For $x Step $2$: Evaluate $(fog)(2)$ and $(gof)(2)$.$(fog)(2) = f(g(2)) = f(3(2)) = f(6) = 6$.$(gof)(2) = g(f(2)) = g(2) = 3(2) = 6$.Step $3$: Evaluate $(fog)(-2)$ and $(gof)(-2)$.$(fog)(-2) = f(g(-2)) = f(-2) = 3(-2) = -6$.$(gof)(-2) = g(f(-2)) = g(3(-2)) = g(-6) = -6$.Step $4$: Check the options.Option $(B)$: $(fog)(2) - (gof)(-2) = 6 - (-6) = 12 
eq 0$.Option $(C)$: $(fog)(2) - (fog)(-2) = 6 - (-6) = 12 
eq 0$.Option $(D)$: $(gof)(2) + (gof)(-2) = 6 + (-6) = 0$. Thus, option $(D)$ is correct.

If $f : R \to R$ and $g : R \to R$ are defined as $f(x) = 2x - |x|$ and $g(x) = 2x + |x|$, then

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