# Finding the inverse of a function

• nowayjose

## Homework Statement

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?

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

## Homework Statement

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?

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