I have read various responses to finding an inverse function in the pre-calculus section and decided to post this question here instead since it isn’t really a ‘pre-calculus’ type of question.. The solutions to finding an inverse of a function look reasonably straight forward but I was wondering whether anyone might be able to assist me in finding an inverse function for the following;

$\displaystyle y= ax + bx^c$

Clearly it is not possible to find 'x' in terms of 'y' since the equation is implicit, hence the format for the explicit examples I've seen can't be used, so how does one approach finding the inverse of this function? Note that I have found the variables a, b, and c and they are not integers for the purposes of my investigation.

Now, having stated the initial problem, it gets a little more complicated as I also have to find the integral to this inverse function but just cannot get past the first step using the examples already provided in the forum. Any helpful tips to get me started would be greatly appreciated.

In order to lay out the problem completely I have presented it in the terms I am presently using.;

If $\displaystyle \tau = \left(\frac{\delta Pr}{2L} \right) = a\gamma+b\gamma^c$

and

$\displaystyle v=\frac{1}{R^2}\int_{0}^R r^2f^{-1}\left(\frac{\delta Pr}{2L} \right)dr$

Then I imagine that by guessing a value for $\displaystyle \delta P$ I can use an iterative process can find a value for $\displaystyle v$ and compare that to known values for v in order to modify my initial guess. I already have defined parameters for $\displaystyle L$ and $\displaystyle R$

I am sure you can now see why I need to find the inverse ! What do you think?

For now if you've got any ideas on how to approach this little problem, I'd be pleased to know.

Cheers!