# Math Help - Odd extension, even extension?

1. ## Odd extension, even extension?

Let the function $f(x) = x^2 + x + \sin x - \cos x + \ln(1 + |x|)$ be defined over the interval [0,1]. Find the odd and even extension of f(x) in the interval $[-1, 1]$

Spoiler:
$\mbox{Odd extension:}\ -x^2 + x + \sin x + \cos x - \ln(1 + |x|)$ $\
\mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)$

How to do it?

2. Originally Posted by fardeen_gen
Let the function $f(x) = x^2 + x + \sin x - \cos x + \ln(1 + |x|)$ be defined over the interval [0,1]. Find the odd and even extension of f(x) in the interval $[-1, 1]$

Spoiler:
$\mbox{Odd extension:}\ -x^2 + x + \sin x + \cos x - \ln(1 + |x|)$ $\$
Spoiler:
$
\mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)" alt="
\mbox{Even extension:}\ x^2 - x - \sin x - \cos x + \ln(1 + |x|)" />

How to do it?
The odd extension $f(x)$ on $[-1,0)$ satisfies:

$g(x)=-f(-x)$

and the even extension:

$h(x)=f(-x).$

CB