Function/1+ln x -1 over x/Bijection/Exercise

We consider the function

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a) Prove that ${}f$ gives a continuous bijection between ${}\mathbb {R} _{+}$ and ${}\mathbb {R}$.

b) Determine the inverse image ${}u$ of ${}0$ under ${}f$, then compute ${}f'(u)$ and ${}(f^{-1})'(0)$. Draw a rough sketch for the inverse function ${}f^{-1}$.

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