6

How can I find the inverse function of $$f(x)=x+\ln(x).$$ This function has an inverse function (I can prove it) but I couldn't find it. Help please!

egreg
  • 238,574
user161440
  • 313
  • Start by identifying the domain and range of $f(x)$. What do you expect to find in the way of an inverse of $f(x)$? – hardmath Jul 14 '14 at 12:30
  • 4
    The inverse function of $f$ is just that; you can't express it in terms of elementary functions. – egreg Jul 14 '14 at 12:34

1 Answers1

13

$y=x+lnx$

$y=ln(e^x)+ln(x)$

$y=ln(xe^x)$

$e^y=xe^x$

Here, you can use the Lambert W Function

$x=W(e^y)$

Thanos Darkadakis
  • 1,869