I have been wondering... Is there a mathematical equation for the inverse of a function? I mean apart from the typical "replace the x's with y's" way... I tried using the inverse function derivative and integral but only got a circular answer! If this isn't possible, what about Taylor series or some other sort of series? Note, the answers shouldn't be circular, meaning the definition of the inverse shouldn't include the inverse...
(Also, please tell me what is wrong with my question or whether there was a duplicate I didn't find so I can fix this question or delete it.)
In the very particular case of an analytic function, there is a relatively explicit formula for the inverse. See Lagrange inversion theorem.
The general formula is: for $f(w)=z$ with $f'(a)\ne 0$, $$f^{-1}(z)=a+\sum_{n=1}^{\infty}\left(\lim_{w\to a}\frac{d^{n-1}}{dw^{n-1}}\left(\frac{w-a}{f(w)-f(a)}\right)^n\right)\frac{(z-f(a))^n}{n!}.$$ In the special case $f(w)=w/\phi(w)$, can be reduced to the easier form $$[z^n]f^{-1}(z)=\frac1n[w^{n-1}]\phi(w)^n$$ ($[\cdot]$ is the coefficient of the corresponding power)
A very interesting example is the deduction of the coefficients of Lambert function, defined by $z=w/e^w$: $$[z^n]w=\frac1n[w^{n-1}]e^{nw}=\frac1n\sum_{k=0}^\infty\frac{(nw)^k}{k!}=\frac{n^{n-1}}{n!}.$$
For several proofs, see Lagrange Inversion Formula by Peter Magyar, Proof of the Lagrange Inversion Formula by Jason Z. Gao, or (maybe paywalled) An algebraic proof of the Lagrange-Bürmann formula by Peter Henrici.