# Inverse of functions

## Main Question or Discussion Point

Let,s suppose we want to obtain the inverse of the functions:

$$y=\frac{sin(x)}{x}$$ $$y=cos(x)+x$$ or $$y=\int_{c}^{x}dt/logt$$

as you can check you can,t explicitly get g(y)=x from y=f(x)..then how would you manage to get it?..i have heard about Lagrange inverse series theorem to invert a series..but what happens if the function is not analytic on the whole real line?..for example includes terms in the form |x|, lnx, 1/x or x^{r} with r a real number.

HallsofIvy
Homework Helper
Most of the functions you mention do not have inverses defined for all real numbers. How are you restricting them so that they do?

you can use some sorts of integrals to find inverse functions.
1)
$$y'=\frac{xcosx-sinx}{x^2}$$
let g be the inverse function,
$$\frac{dg}{dx}=\frac{1}{y'(y)}$$
substitute $$y=\frac{sinx}{x}$$

integrate it and get g..... well its gonna be messy. however, as hallsofivy said, these functions have restrictions... so be careful.

matt grime
$$y=\frac{sin(x)}{x}$$ $$y=cos(x)+x$$ or $$y=\int_{c}^{x}dt/logt$$