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

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The function f(x) = x/(1+|x|) has a range of (-1, 1). By analyzing the function separately for positive and negative values of x, it is established that as x approaches positive or negative infinity, f(x) approaches 1 and -1, respectively. The function is continuous and bounded, confirming that the image is indeed the interval (-1, 1). Elementary theorems of analysis support this conclusion, providing a solid foundation for understanding the behavior of the function.

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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 !
 
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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.
 

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