# What is the inverse function of f(x) =5x^3 + 4x^2 + 3x + 4?

Jan 2, 2016

$x = 5 {y}^{3} + 4 {y}^{2} + 3 y + 4$

#### Explanation:

The simplest way to write the inverse function is to write $y$ in place of $f \left(x\right)$ and switch all the $x$s with $y$s and the $y$ with $x$, giving:

$x = 5 {y}^{3} + 4 {y}^{2} + 3 y + 4$

This can be written explicitly (in terms of $y$), but it uses the incredibly cumbersome cubic formula and is overall unhelpful. But, for the sake of interest, it's as follows:

$x = - {\left(15 \sqrt{3} \sqrt{675 {y}^{2} - 4576 y + 7900} - 675 y + 2288\right)}^{\frac{1}{3}} / \left(15 \left({2}^{\frac{1}{3}}\right)\right) + \frac{29 \left({2}^{\frac{1}{3}}\right)}{15 {\left(15 \sqrt{3} \sqrt{675 {y}^{2} - 4576 y + 7900} - 675 y + 2288\right)}^{\frac{1}{3}}} - \frac{4}{15}$

Graph: