# Solving for an equation in the form ax^3 +cx + d = y

#### OscillationContinuum

##### New member
Thank God this website has LaTeX haha.

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?

#### Subhotosh Khan

##### Super Moderator
Staff member
Thank God this website has LaTeX haha.

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?
Is That the actual problem or "a part" of some other problem.

#### Dr.Peterson

##### Elite Member
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?