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

Question:

If $f: R \rightarrow R$ is given by $f(x)=2 x+|x|$, then $f(2 x)+f(-x)+4 x=$  _______________.

Solution:

Given: f(x) = 2x + |x|

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

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

$\Rightarrow f(2 x)=4 x+2|x|$  $\ldots(1)$

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

$\Rightarrow f(-x)=-2 x+|x|$  $\ldots(2)$

Now,

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

$=6 x+3|x|$

$=3(2 x+|x|)$

$=3 f(x)$

Hence, $f(2 x)+f(-x)+4 x=\underline{3 f(x)}$.

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