Question

Let f : R → R be defined by f(x) = 2x + |x|. Then f(2x) + f(−x) − f(x) =

(a) 2x
(b) 2|x|
(c) −2x
(d) −2|x|

Answer 1

(b) 2|x|

f(x) = 2x + |x|
Then, f(2x) + f(−x) − f(x)
$$\begin{gathered} =2\left(2x\right)+2\left|x\right|+\left(-2x\right)+\left|-x\right|-2x+\left|x\right| \\ =4x-2x-2x+2\left|x\right|+\left|-x\right|-\left|x\right| \\ =0+2\left|x\right|+\left|x\right|-\left|x\right| \\ =2\left|x\right| \end{gathered}$$