What is the inverse function of e^x+x? As the natural log(x) function is the inverse of the exponential e^x and log(x+1) is the inverse of e^x−1, what it the inverse of e^x+x?

Ryder Ferguson

Ryder Ferguson

Answered question

2022-10-15

What is the inverse function of e x + x
As the natural log ( x ) function is the inverse of the exponential e x and log ( x + 1 ) is the inverse of e x 1, what it the inverse of e x + x?

Answer & Explanation

erkvisin7s

erkvisin7s

Beginner2022-10-16Added 12 answers

That inverse function cannot be expressed by the usual elementary functions, but Wolfram Alpha finds it to be
f 1 ( y ) = y W ( e y )
where W is Lambert's W function.
This at least gives it a name, but may not otherwise seem like much progress because the W function is itself defined as the inverse of x e x . So in some sense all we have achieved is relate the solution to your problem to the solution of another similarly difficult problem ...

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