interchanging the variables in the equation

Question

I am given the following equation $$y=7.515x^3-10.229x^2+5.05x$$ How do I find the value of $x$ when $y$ is given. I need to somehow interchange the position of $x$ and $y$ so that the equation has only $x$ on left side of equality.

Answer 1

The polynomial function $f(x) =7.515 x^3 - 10.229 x^2 + 5.05 x$ has the derivative $f'(x) = 22.545 x^2 - 20.458 x + 5.05$, which is positive for all $x \in \mathbb{R}$ with a local minimum at $x_m=20.458/45.090$ with $f'(x_m) \approx 0.887451$.

Therefore the inverse function $f^{-1} : \mathbb{R} \rightarrow \mathbb{R}$ is well-defined for all $x \in \mathbb{R}.$ So to get $x$ given $y$ you have to compute $x = f^{-1}(y)$, i.e. compute the real root of the cubic equation $$7.515 x^3 - 10.229 x^2 + 5.05 x - y = 0.$$ This can be done with the formulas given in the Wikipedia article or with a root-finding algorithm like Newton's method or with bisection.