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: Understanding the Concept
\(f(x)=x\) for \(x\ge0\) and \(3x\) for \(x<0\). \(g(x)=3x\) for \(x\ge0\) and \(x\) for \(x<0\).

Step 2: Key Formula or Approach
Evaluate \(f\circ g\) and \(g\circ f\) at \(\pm2\).

Step 3: Detailed Explanation
\((f\circ g)(2)=f(6)=6\). \((g\circ f)(2)=g(2)=6\).
\((f\circ g)(-2)=f(-2)=-6\). \((g\circ f)(-2)=g(-6)=-6\).
(A) \(6+6=12\neq0\). (B) \(6-(-6)=12\neq0\). (C) \(6-(-6)=12\neq0\).
(D) \((g\circ f)(2)+(g\circ f)(-2)=6-6=0\), which is true.

Final Answer:
Statement (D) holds, option (D). \[ \boxed{\text{(D)}} \]