Question

If f : R → R is given by f(x) = 2x + |x|, then f(2x) + f(–x) + 4x = _______________.

Answer 1

Given: f(x) = 2x + |x|

$$\begin{gathered} f\left(x\right)=2x+\left|x\right| \\ f\left(2x\right)=2\left(2x\right)+\left|2x\right| \\ \Rightarrow f\left(2x\right)=4x+2\left|x\right|...\left(1\right) \\ f\left(-x\right)=2\left(-x\right)+\left|-x\right| \\ \Rightarrow f\left(-x\right)=-2x+\left|x\right|...\left(2\right) \\ \mathrm{Now}, \\ f\left(2x\right)+f\left(-x\right)+4x=4x+2\left|x\right|-2x+\left|x\right|+4x \\ =6x+3\left|x\right| \\ =3\left(2x+\left|x\right|\right) \\ =3f\left(x\right) \end{gathered}$$

Hence, f(2x) + f(–x) + 4x = 3f(x).