# Inverse Function

1. Jan 22, 2010

### shiri

I got a problem for this question.

Given the function y=f(x)=6x^3+6x+2. Find f^-1(x).

Can anybody tell me how to solve this function?

So far I got this:

x=6y^3+6y+2

x-2=6y^3+6y

I just couldn't solve for y variable. So, please tell what I do wrong?

thanks

2. Jan 22, 2010

### CompuChip

It doesn't look like you are doing anything wrong.
Just note that it is a cubic polynomial. So although
x-2 = 6y3+6y
will definitely have solutions for y(x), they may be non-trivial to write down.

However, a solution exists.

3. Jan 22, 2010

### Altabeh

You couldn't because it is not that much easy. I'm going to sort some steps that you should carefully follow to get x in terms of y:

1- Define two new variables u and v, with uv = 2 and x=u-v. Introduce this into your function and simplify the result.

2- Put v=2/u in the new function.

3- Now you have a simple function which is quadratic in u^3. From there, I assume, you know how to solve a quadratic equation for t=u^3 after taking y to the right side and letting one side just be lonely with a zero!

4- Now from 1,2 and 3 we have $$x=u-2/u = t^{1/3}-2t^{-1/3}$$ and this is what you want since t is a function of y.

AB