Thread: Proving that a function is odd

1. Proving that a function is odd

Hey math forum folks,

I know that f(x) = sinx is an odd function. With this knowledge, I therefore know that f(x) = sinx + x is also an odd function. However, I want to be able to prove that and I'm not sure how because I face the challenge with having to add sin pi/2 + pi/2 which I have no clue how.

I attached an image to what I have done so far (I apologize for the messy appearance).

I hope someone can help!

- Olivia

2. Re: Proving that a function is odd

if $f(-x) = -f(x)$ the function is odd ...

$f(x) = \sin{x} + x$

$f(-x) = \sin(-x) + (-x) = -\sin{x} - x = -(\sin{x}+x) = -f(x)$

note ...

$\sin(-x) = \sin(0-x) = \sin(0)\cos(x) - \cos(0)\sin(x) = 0 \cdot \cos(x) - (1) \cdot \sin(x) = -\sin(x)$

3. Re: Proving that a function is odd

$k(x)\ is\ an\ odd\ function \iff k(x) = -\ k(-\ x).$ Definition.

$k(x) = -\ k(-\ x) \iff -\ k(x) = k(-\ x).$ Simply multiply equation by - 1.

$\therefore k(x)\ is\ an\ odd\ function \iff -\ k(x) = k(-\ x).$

$Let\ f(x)\ and\ g(x)\ be\ odd\ functions\ and\ h(x) = f(x) + g(x).$

$h(-\ x) = f(-\ x) + g(-\ x) = -\ f(x) - g(x) = -\ \{f(x) + g(x)\} = -\ h(x).$

$\therefore h(x)\ is\ an\ odd\ function.$