I'm trying to solve (b) but while I'm stuck at $y^2+2y+1-yx+x=0$ .
Even I have a proper inverse with the following structure: $y=$ something, would that be enough to prove the $f$ has an inverse or is there more work to do?
[Math] how to prove that a function has an inverse
algebra-precalculus
Best Answer
Note that you have to show that the inverse of $f$ exists, not to find it explicitly.
By looking at the derivative it should be easy to show that $f$ is strictly decreasing which means that is $f$ is injective. Are you able to show that $f$ maps $(0,+\infty)$ onto $(0,+\infty)$? Use the continuity of $f$. This will imply that $f$ is a bijection from $(0,+\infty)$ to $(0,+\infty)$ and therefore $f$ has an inverse map $g$.
As regards the derivative of the inverse, note that $f(1)=\frac{4}{4+\pi}$ and take a look at the inverse function theorem.