inverse functions

[arbagby2010]

if g(x)=5+x+e^x find g^-1(6)
x=5+y+e^y
x-5=y+e^y
ln(x-5)=ln(y+e^y)
ln(x-5)=lny+y
what is next?

 

[daon]

As g is a 1-1 function, you need to solve g(x) = 6, i.e. e^x=1-x, which by observation, happens at x=0.

Algebraically, it is not so straight forward, but with the use of the series definition of e^x:

\(\displaystyle e^x +x -1 = \sum_{n =1}^{\infty} x^n/n! + x -1 = 1+\sum_{n =2}^{\infty} x^n/n! + x -1 = x \left (1+\sum_{n =2}^{\infty} x^{n-1}/n! \right )\)