Question: Given f(x)=e^{x+4}-1, x \in R. Find f^{-1}(x).
1. **State the problem:** We are given the function $$f(x) = e^{x+4} - 1$$ and need to find its inverse function $$f^{-1}(x)$$.
2. **Recall the definition of inverse function:** The inverse function $$f^{-1}(x)$$ satisfies $$f(f^{-1}(x)) = x$$ and $$f^{-1}(f(x)) = x$$.
3. **Set up the equation:** Let $$y = f(x) = e^{x+4} - 1$$. To find $$f^{-1}(x)$$, we solve for $$x$$ in terms of $$y$$:
$$y = e^{x+4} - 1$$
4. **Isolate the exponential term:**
$$y + 1 = e^{x+4}$$
5. **Take the natural logarithm on both sides:**
$$\ln(y + 1) = x + 4$$
6. **Solve for $$x$$:**
$$x = \ln(y + 1) - 4$$
7. **Replace $$y$$ with $$x$$ to express the inverse function:**
$$f^{-1}(x) = \ln(x + 1) - 4$$
**Note:** The user-provided answer $$f^{-1}(x) = \ln(x + 4) + 1$$ is incorrect based on the original function.
**Final answer:**
$$f^{-1}(x) = \ln(x + 1) - 4$$