I have been working on a project where I have a function F(x) which is closely related to a function G(x). The function G(x) has been well studied already and tools exist for using it. To make the equations I derive for F(x) useful I wanted to convert them into G(x) so that people can evaluate things for F(x) using the tools that already exist for G(x)

So far I derived their relationship

$\displaystyle r\cdot F(x)+ c= G(x)$

So for example if I had this equation to find an important number B

$\displaystyle B= F(5)-F(y)$

I could turn it into

$\displaystyle B= \frac{G(5)-c}{r}-\frac{G(y)-c}{r}$

$\displaystyle B= \frac{G(5)-G(y)}{r}$

And use the tools that exist for G(x) to find B for any y

But I am stuck when it comes to their inverse functions. As an example of this I have derived the equation

$\displaystyle L=F^{-1}(y-0.5)$

I am confused about how I can use the relationship $\displaystyle r\cdot F(x)+ c= G(x)$ to get L in terms of the inverse function of $\displaystyle G$. I am not even sure if this is possible to do