Question

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

• A. \((fog)(2)+(gof)(2) = 0\)
• B. \((fog)(2)-(gof)(-2) = 0\)
• C. \((fog)(2)-(fog)(-2) = 0\)
• D. \((gof)(2)+(gof)(-2) = 0\)

Hint:

Write \(f\) and \(g\) piecewise, then evaluate the compositions.

Answer 1

The Correct Option is D

Step 1: Simplify the compositions
For $x>0$: $g(f(x))=g(x)=3x$. For $x<0$: $g(f(x))=g(3x)=3x$. So $g\circ f(x)=3x$ for all $x$.

Step 2: Check (D)
$(g\circ f)(2)+(g\circ f)(-2)=6+(-6)=0$, so option (D) is true.

Final Answer:
Only (D) gives zero, option (D). \[ \boxed{\text{(D)}} \]