What is the inverse of the following function

Question

what is the inverse of

$$G(x):=\exp(-\exp(-x))?$$

Answer 1

$$y=\exp(-\exp(-x))$$ $$\ln(y)=-\exp(-x)$$ $$-\ln(y)=\exp(-x)$$ $$\ln(-\ln(y))=-x$$ $$-\ln(-\ln(y))=x$$

So the inverse function is $$G^{-1}(x)=-\ln(-\ln(x))$$

Answer 2

Let $y=e^{-e^{-x}}$ then $\ln y=-e^{-x}\implies -\ln y= e^{-x}\implies \ln((-\ln y))=-x\implies x=-\ln((-\ln y))$