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

We consider the function

$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=1+\ln x-{\frac {1}{x}}.$

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 ${}f^{-1}$.

Create a solution