Image of f(x) = x/(1+|x|): Find the Range

[Demonoid]

I've given a function where f: R->R, and need to determine an image(or range):

f(x) = x/(1+|x|)

I've pretty sure the image is R, but I'm not positive:

Heres my attempt:

y/1 = x/(1+|x|)
y(1+|x|) = x
y+ y|x| = x
y|x| = x - y... I'm kinda stuck here, since I can't determine an image from this ?


any help is welcome !

 

[disregardthat]

Split the function up in for positive and negative x. Then it should be easy to find upper and lower bounds. Together with some elementary theorems of analysis it should be easy to argue what the image can be.