Question

Let f: R → R be defined by f(x) = 2x + |x|, then f (2x) + f(-x) -f(x) is equal to *

Answer 1

$</p><p></p><p>\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

By the definition of modulus

[tex] |x| =

\begin{cases}

\: \: \: x& \text{if } x \geqslant 0 \\ \\

- x & \text{if } x < 0

\end{cases}[/tex]

So

$| - x| = |x|$

GIVEN

$Let f: \mathbb{R} → \mathbb{R} \: \: be \: defined \: by \: f(x) = 2x + |x|$

TO EVALUATE

$f (2x) + f(-x) -f(x)$

EVALUATION

$f (2x) = 2 \times 2x + |2x| = 4x + 2 |x|$

$f(-x) = 2 \times ( - x) + | - x| = - 2x + |x|$

$f(x) = 2x + |x|$

Hence

$f (2x) + f(-x) -f(x) = 4x + 2 |x| - 2x + |x| - 2x - |x| = 2 |x|$