How can I establish the inverse function? $$f(x)=x^{4}-3x^{3}+4x^{2}-6x+4$$ redefining the function so that it is bijective I obtained: $$\begin{array}{llll}f:&\left[\dfrac{3}{2},\infty\right)&\longrightarrow&\left[-\dfrac{17}{16},\infty\right)\\&x&\longmapsto&f(x)=x^{4}-3x^{3}+4x^{2}-6x+4\end{array}$$
How can I express the inverse function, or is it not possible?
There is no inverse function to $f(x)=x^{4}-3x^{3}+4x^{2}-6x+4$. This is because $f(x)$ has two $x$ values for one $y$ value, which means $f^{-1}(x)$ will have two $y$ values for one $x$ value. Since this does not pass the vertical line test, the inverse of $f(x)=x^{4}-3x^{3}+4x^{2}-6x+4$ is not a function.
If you want to get the inverse function of $f(x)=x^{4}-3x^{3}+4x^{2}-6x+4$ on a specific domain to make the inverse a function, one way is to use the tedious quartic formula, which restates the inverse function as $f^{-1}(x)$ in terms of $x$.