Can I find the inverse function of the original function by using inverse function theorem?

Question

I learn about the inverse function theorem which says the derivative of the inverse function is the reciprocal of the derivative of the original function, and I wonder if it is possible that I can find the inverse function of the original function by applying this theorem? e.g. If I have $$ f(x) = \frac{2^x}{2^x+1} $$ then I can find $$f^{-1}(x)$$ by $$f^{-1}(x)=\int\frac{dx}{f'(x)}.$$

Answer 1

The inverse function theorem says $$(f^{-1}(x))'=\frac1{f'(f^{-1}(x))}$$ which can be reworked by integration, $$f^{-1}(x)+C=\int\frac{dx}{f'(f^{-1}(x))}.$$ Unfortunately, $f^{-1}$ appears in both members and this is an integral equation, usually much more difficult than the function inversion itself.

Answer 2

It's not likely to be helpful in a way that you're looking for. Take a simpler example, $f(x)=\sin(x)$. Then the inverse function theorem says that, there exists a $g_{x_0}(y)$ an inverse of $f(x)$ near $x_0$, and $g_{x_0}'(f(x))=\frac{1}{f'(x)}=\frac{1}{\cos(x)}$. If this were expressed in terms of $f(x)$, we could now integrate to find $g(y)$, but it's not. The inverse function theorem tells you the inverse as a function of $x$, which is probably not enough for you. In order to start integrating, we already have to answer the question: "okay, for a given $y$ value, what value of $x$ do I use to get the inverse from the inverse function theorem?" In other words, you already need to know the inverse.