# Thread: Odd extension, even extension?

1. ## Odd extension, even extension?

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

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

How to do it?

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

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

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

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

and the even extension:

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

CB