# Finding the inverse of a function

1. May 30, 2012

### nowayjose

1. The problem statement, all variables and given/known data

Find the inverse function of $f(x)= 2x + lnx$

2. The attempt at a solution

So usually i would start out with
$y= 2x+lnx$ (finding x terms of y)
and then switch x and y... but i can't figure out how to find x. doesn't seem to be possible to factorize this... are there other ways of doing this?

2. May 30, 2012

### Whovian

I'm 99% certain this function's inverse isn't expressible in terms of elementary functions.

3. May 30, 2012

### Ray Vickson

The solution cannot be expressed in terms of "elementary" functions. It can be solved in terms of the so-called Lambert W-function: the solution of $y = 2x + \ln(x)$ is
$$x = e^{y - W(2e^y)},$$
where W(v) (Lambert W-function) is defined as the root of the equation $z \:e^z = v$ which is analytic at v = 0.

RGV