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what i did was set [tex] y = \frac{x}{1 + |x|} [/tex] so that [tex] |y| = \frac{|x|}{1 + |x|} [/tex] and solving for |x| i get [tex] |x| = \frac{1}{1 - |y|} [/tex]. but from here i'm not sure about my logic. i argued that since y and x are basically vectors in [tex] \mathbb{R}^n [/tex] pointing in the same direction, given a vector y, i just need to scale it properly so that it becomes x. since i have the length of x is [tex] |x| = \frac{1}{1 - |y|} [/tex], and since |y| is not 1, we should take [tex] \frac{y}{|y|} [/tex] so that the vector is now of length 1 and pointing in the direction of x, and then multiply by the factor [tex] |x| = \frac{1}{1 - |y|} [/tex] so the inverse function would be [tex] g(y) = \frac{y}{|y|(1 - |y|)} [/tex]. but this doesn't work and i am having trouble figuring out why.

can someone help explain what i did wrong?