Can this be represented as $f:x\to y$? Its inverse function?

Question

$$y+5x+6x^4y=0$$

What will be the inverse function of this quartic equation?

Answer 1

$x$ as a function of $y$ is a root of a polynomial of degree $4$.

Here you can find some general methods to compute those roots.

Note: Be aware, that if you are studying calculus, there are many properties of inverse functions/ implicit functions that can be determined without computing the inverse.

Answer 2

Yes it can. There are four such functions $y:x\mapsto y(x)$, each one defined differently according to the four possible solutions to your equation. The closed forms of these functions are quite nasty, check this: https://www.wolframalpha.com/input/?i=isolate(y%2B5x%2B6x4y%3D0,y)

What you can do if you want to study the behavior of these functions rather than having closed forms is to take implicit derivatives and study them.

Answer 3

Note that since

• $y(0)=0$ and $x\to \infty \quad y(x)\to 0$

by EVT $y(x)$ is not injective and thus it is not invertible over the whole domain.

The inverse exist for those restrictions with $y(x):[a,b]\to[c,d]$ bijective.