Find \( f(2x)+f(-x)-f(x) \)
Question:
Let \( f:R\to R \) be defined by
\[ f(x)=2x+|x| \]
Then
\[ f(2x)+f(-x)-f(x) \]
is equal to
(a) \(2x\)
(b) \(2|x|\)
(c) \(-2x\)
(d) \(-2|x|\)
Solution:
\[ f(2x)=4x+|2x| =4x+2|x| \]
\[ f(-x)=-2x+|-x| =-2x+|x| \]
\[ f(x)=2x+|x| \]
Therefore,
\[ f(2x)+f(-x)-f(x) \]
\[ =(4x+2|x|)+(-2x+|x|)-(2x+|x|) \]
\[ =2|x| \]
\[ \boxed{\text{Correct Answer: (b)}} \]