1. The problem is to find the inverse function $f^{-1}(x)$ of the function $f(x) = 1 + e^{x}$. 2. To find the inverse, we start by setting $y = f(x)$, so: $$y = 1 + e^{x}$$ 3. The goal is to solve for $x$ in terms of $y$. 4. Subtract 1 from both sides: $$y - 1 = e^{x}$$ 5. Take the natural logarithm (ln) of both sides to undo the exponential: $$\ln(y - 1) = \ln(e^{x}) = x$$ 6. Therefore, the inverse function is: $$f^{-1}(x) = \ln(x - 1)$$ 7. Important note: The domain of $f^{-1}(x)$ is $x > 1$ because the argument of the logarithm must be positive. Final answer: $$f^{-1}(x) = \ln(x - 1)$$