Question

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

Options

• (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)

\[= 2\left( 2x \right) + 2\left| x \right| + \left( - 2x \right) + \left| - x \right| - 2x + \left| x \right|\]
\[ = 4x - 2x - 2x + 2\left| x \right| + \left| - x \right| - \left| x \right|\]
\[ = 0 + 2\left| x \right| + \left| x \right| - \left| x \right|\]
\[ = 2\left| x \right|\]