Anyway, I was tasked with finding the inverse function of

\(\displaystyle f(x) = x^{3} + 3x + 4\). But I'm stuck... Please can someone explain where to go after switching f(x) and x?

In general, a cubic can't be solved algebraically without some very complicated work. Graphing this one, I see that it is invertible (one-to-one), and that it is equal to \(\displaystyle (x+1)(x^2-x+4)\). But this doesn't help in inverting it algebraically. There is a "cubic formula", but it is ugly.

You have to solve \(\displaystyle y^3+3y+(4-x) = 0\), for y. This is a depressed cubic equation, which is simpler than the general case; you can see the formula for the solution

here. It still isn't very nice.

Who gave you this task, and how was it worded? What is the actual goal?