Having trouble finding the inverse of the function f(x)=x^5 + x^3 + x

3 Answers

Only polynomials of degree 1 through 4 have general formulas for their inverses (real and complex). Quintics and higher do not have a general formula for the inverse, so some polynomial inverses can be impossible to solve in terms of radicals even if it exists.
The inverse exists, but it's unlikely that there's any simple expression for it.
Interesting function, it actually factors nicely as x(x+cis(pi/6))(x-cis(pi/6)(x+cis(-pi/6)(x-cis(-pi/6), where cis is a way to describe complex numbers.

~~This is a little bigbrain but one thing you could try is using the fact that the derivative of the inverse is the reciprocal of the derivative. Then the derivative of f-inverse is 1/(5x^4 + 3x^2 + 1), and we can find f-inverse by integrating via partial fraction decomp and using the initial condition that f(0)=0.~~

~~Plugging this into an integral calculator spits out an absolute monstrosity involving roots inside of logs and arctans, but it is doable because of the nice geometry of the derivative (quadratic in x^2).~~

Edit: Nevermind, this is NOT how that rule works.

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