Why do we move the minus sign in front of sin in odd/even function equations?

[braceman]

Hi all,

Doing some odd/even functions questions and I've had a bit of a weird mind fart when trying to remember why we process the equations like we do...It's probably real simple, but I just can't remember why we 'move' one of the signs to where it ends up...please bear with me (and don't laugh too much)...

While determining if f(x) = (x^2 + 2)sin(x) = odd/even

f(-x) = (-x^2 + 2)sin(-x)

(-x^2 = x^2 and sin(-x) = -sin(x)

therefore the next line should theoretically be

f(-x) = (x^2 + 2)(-sin(x))

but (after checking on wolfram and it's giving it as an odd function) I'm sure we need it to be =

f(-x) = - (x^2 + 2)sin(x)

thus making f(-x) = -f(x) and hence odd

but for the life of me, I can't remember why we move the - sign infront of 'sin' to the beginning, making it - (x^2 + 2).
Is it just for 'ease' or is there a specific reason? I can remember seeing it before, but like I said, I just can't remember 'why'.

Any simple explanations out there would be appreciated...

 

[arildno]

f(-x) = (x^2 + 2)(-sin(x))=(x^2 + 2)*((-1)*sin(x))=(-1)*((x^2 + 2)*sin(x))=-(x^2 + 2)*sin(x)

 

[HallsofIvy]

You are probably entering the function into Wolfram incorrectly. (-x)^2 is NOT the same as -x^2!

If x= 4, (-x)^2= (-4)^2= 16 but -x^2= -(4^2)= -16.