How to prove or disprove the inverse $\displaystyle x=f^{-1}(y)$ of the rational function $\displaystyle y=f(x)$ is also a rational function?

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- Sep 12th 2010, 03:45 AMHinaticoInverse of rational function
How to prove or disprove the inverse $\displaystyle x=f^{-1}(y)$ of the rational function $\displaystyle y=f(x)$ is also a rational function?

- Sep 12th 2010, 05:33 AMchisigma
A simple example: $\displaystyle y(x)= x^{3} + x$ is rational but the inverse $\displaystyle x(y)$ is irrational. To prove that is sufficient to consider that the real solution of the equation $\displaystyle x^{3} + x -y =0$ is obtained using the Cardano's formula, where square and cubic roots are involved...

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$\displaystyle \chi$ $\displaystyle \sigma$