Math Help - Finding inverse using hyperbolic function

1. Finding inverse using hyperbolic function

I'm getting spoonfed instructions and I still don't get it. >_>

I am to find the inverse of f(x) = x^3 + x, by using properties of sinh(x).
The question tells me to let y = x^3 + x, let z = $\frac{\sqrt{3}}{2}x$ and then prove that $\frac{3\sqrt{3}}{2}y = 4z^3 + 3z$.

After I do that, it tells me to set z = sinh(θ), solve for θ and then reverse the changes to the variables to find x as a function of y.

I managed to do most of it, up until the 'reverse the changes' part. My θ equals something ridiculously long and confusing, and I have no idea how to use it to my benefit.
$\theta = \frac{sinh^{-1}(\frac{3\sqrt{3}}{2}y)}{3}$

Just need a bit of guiding light, it feels like I overlooked something...

EDIT: All right, well, I solved for θ in terms of x now too.
$\theta = sinh^{-1}(\frac{\sqrt{3}}{2}x)$

And then I equate the two.
$sinh^{-1}(\frac{\sqrt{3}}{2}x) = \frac{sinh^{-1}(\frac{3\sqrt{3}}{2}y)}{3}$

And I'm not exactly sure how to solve...

2. Originally Posted by BlackBlaze
I'm getting spoonfed instructions and I still don't get it. >_>

I am to find the inverse of f(x) = x^3 + x, by using properties of sinh(x).
The question tells me to let y = x^3 + x, let z = $\frac{3\sqrt{3}}{2}x$ and then prove that $\frac{3\sqrt{3}}{2}y = 4z^3 + 3z$.

After I do that, it tells me to set z = sinh(θ), solve for θ and then reverse the changes to the variables to find x as a function of y.

I managed to do most of it, up until the 'reverse the changes' part. My θ equals something ridiculously long and confusing, and I have no idea how to use it to my benefit.
$\theta = \frac{sinh^{-1}(\frac{3\sqrt{3}}{2}y)}{3}$

Just need a bit of guiding light, it feels like I overlooked something...

EDIT: All right, well, I solved for θ in terms of x now too.
$\theta = sinh^{-1}(\frac{\sqrt{3}}{2}x)$

And then I equate the two.
$sinh^{-1}(\frac{\sqrt{3}}{2}x) = \frac{sinh^{-1}(\frac{3\sqrt{3}}{2}y)}{3}$

And I'm not exactly sure how to solve...
If $y = x^3 + x$ and $z = \frac{3\sqrt{3}}{2}x$ then $\frac{3\sqrt{3}}{2}y = 4z^3 + 3z$ does not seem correct to me.

3. Right you are, it has been acknowledged.

However, I managed to complete the question. I first isolated for y in the 'θ in terms of x = θ in terms of y' equation. Then I swapped around the x and y and isolated for y a second time.

Rather crude, but it got the job done.